## Abstract

This paper deals with control and anti-control of overturning of a rigid block subjected to a generic periodic excitation. Attention is focused on two relevant thresholds, corresponding to heteroclinic bifurcation and immediate overturning, and representing lower and upper bounds of the region where toppling can occur. The two opposite problems of increasing (control) or decreasing (anti-control) of these two curves by properly modifying the shape of the excitation are investigated in depth and the optimal excitations permitting their maximum variations are determined. The notions of ‘global’ and ‘one-side’ control (anti-control) are utilized and their different importance for the various cases is discussed. The effects of control (anti-control) of one curve on the uncontrolled (non-anti-controlled) curve are also investigated, both analytically and with numerical overturning charts. A good agreement is seen to occur.

## 1. Introduction

Control and anti-control of nonlinear dynamics and chaos are new research frontiers in applied dynamical systems theory. In fact, the quite recent ‘control of chaos’ theory, started conventionally by the celebrated work of Ott *et al*. (1990) (although other works appeared at about the same time, e.g. Lima & Pettini (1990), Shaw (1990)), was rapidly paralleled by the ‘anti-control of chaos’ theory (Chen 1998), sometimes also called ‘chaotification’ (Wang & Chen 2000). In the early works, authors specifically refer to ‘chaos’, but it is now understood that this is somehow restrictive, as usually the developed methods have a wider range of application and meaning, so that referring to ‘control and anti-control of nonlinear dynamics’ appears more exhaustive.

Control of chaos is an extremely prolific research area. Among the various books (Chen & Dong 1998) and surveys (Blazejczyk *et al*. 1993; Lindner & Ditto 1995), we mainly refer to Fradkov's ‘Chaos Control Bibliography (1997–2000)’ (2000), which quotes about 700 references on this matter. The original seminal idea is that of *exploiting* the chaotic behaviour of systems to control their dynamics (Ott *et al*. 1990; Lima & Pettini 1990; Chacón & Díaz Bejarano 1993; Chacon *et al*. 2003; Lenci & Rega 1998*a*), thus passing from analysis to synthesis of chaos. Later on, several works merely aimed at removing chaos, by means of classical control techniques (Sifakis & Elliott 2000), empirical methods (Singer *et al*. 1991) or other keen approaches (Pyragas 1992) have also been referred to as ‘control of chaos’, though in this case some authors prefer the name ‘suppression of chaos’ (Lindner & Ditto 1995), which focuses on the effects of control rather than on the underlying skill of the control method. Apart from this general distinction, yet there is no exhaustive classification of the matter, probably because of its young settling down as a research field and of the many developed tools and techniques, although some attempts have recently been made (Chen & Dong 1998; Fradkov 2000).

In turn, the idea of anti-control is very simple, namely using the same trick developed to avoid chaos in order to enhance it. While control or elimination of chaos has a clear practical meaning, at a first glance it is more surprising that anti-control has its own interest, too; yet, ‘… when chaos is under control, it provides the designer with an exciting variety of properties, richness of flexibility, and a cornucopia of opportunities…’ (Chen 1998). Anti-control has been applied to secure communications (Nan *et al*. 2000), biological systems (Christini & Collins 1995), artificial neural network (Bondarenko 2002), maps (Chen & Lai 1998), etc. and has been investigated from both experimental (Moon *et al*. 2003) and purely theoretical (Li & Chen 2003) points of view. In the mechanical context, chaos may be useful in liquid mixing, spreading modal energy at resonance, metal cutting and surface polishing, breaking static friction, targeting and related problems (e.g. to push a mass in a desired potential well).

Anti-control can also be used for a safe design against, e.g. earthquake or unknown excitations. The investigation of this point is one achievement of the present work, where control and anti-control of nonlinear dynamics are applied to a rigid block on a vibrating foundation. This is an old and appealing mechanical problem dating back to Milne (1881) and Perry (1881), although one of the basic references is the more recent work of Housner (1963). In spite of its apparent simplicity, this problem deserves both a practical and a theoretical intrinsic mechanical interest.

It was originally introduced for estimating magnitudes of earthquakes from observations of the response of columns to seismic excitations, and was later dealt with to investigate the stability under ground motion of free-standing structures, such as water towers, nuclear fuel rods, ancient monuments, furniture in civil apartments, etc. On the other hand, the theoretical interest stands in the growing attention towards non-smooth applications exhibiting nonlinear features with no counterpart in smooth systems.

To focus attention on the main features of control and anti-control of nonlinear dynamics and chaos without redundancies, two major simplifications, commonly used in the literature, are employed: (i) the so-called Housner model (1963), according to which the block can only rock without sliding and up-lifting and (ii) piece-wise linear equations of motion. The former is accurate if the block–foundation dry friction is sufficiently high or the block is inserted in a nick, and the block is heavy enough; the latter are accurate for slender blocks.

Among various mechanical features of the rigid block, we consider (i.e. control and anti-control) the overturning of the rest position, which motivated the early works and appears to be the most important practical event. In the case of harmonic excitation, this question has also been investigated in Lenci & Rega (2003*a*).

Among various control techniques, we choose to modify the shape of the periodic excitation. In the case of control, this means, for example, adding controlling superharmonics to a reference harmonic excitation; while in the case of anti-control, this entails assuming the excitation to be periodic and looking for the ‘worst’, in appropriate sense, excitation shape and phase. This technique has been recently investigated in depth by the authors in a series of works (Lenci & Rega 1998*a*,*b*, 2003*b*, 2005) aimed at controlling nonlinear dynamics and chaos of various mechanical systems and models. These works are herein extended (i) by considering not only homo/heteroclinic bifurcation thresholds but also another, mechanically meaningful, threshold and (ii) by investigating, for the first time, the issue of anti-control.

Another element of novelty of this paper is the use of generic periodic excitations, which add to the impulsive (Housner 1963), deterministic harmonic (Hogan 1989; Capecchi *et al*. 1996; Fielder *et al*. 1997) and stochastic (Lin & Yim 1996) excitations previously considered in the literature. Here, the shape of the excitation is chosen as the result of an optimization process, but the investigation of system response under other (non-optimal) non-harmonic excitations is in any case of interest.

Finally, we point out that the proposed control and anti-control techniques, and their relationships, are very general and *not* specific to the rocking block, which in this respect is herein considered just as an archetype model. The generality of the control method has already been shown in a previous authors' work (Lenci & Rega 2004), and it is conjectured that it also holds with respect to anti-control.

## 2. Equations of motion and overturning

The piece-wise linear dimensionless equations of planar motion of the slender homogeneous rigid block rocking around base corners in a constant gravitational field (figure 1) are (Lenci & Rega 2005)(2.1a)(2.1b)(2.1c)Equations (2.1*a*,*b*) describe the rotation around points B and A, respectively, equation (2.1*c*) is the Newton restitution law at the impact, which is supposed to be instantaneous. The angle *φ* characterizes the system state and is measured with respect to the vertical direction, the angle *α* is the block shape parameter, *δ*>0 is the viscous damping, *r*∈[0,1] is the constant coefficient of restitution measuring the dissipation at impacts, and(2.2)is the *generic T*-periodic (*T*=2*π*/*ω*) external excitation representing the dimensionless horizontal acceleration of the rigid foundation (figure 1).

For *δ*=*γ*_{j}=0 and *r*=1, the system is conservative and the associated phase portrait is depicted in figure 2. There exists a unique potential well around the degenerate stable rest position *φ*=0, which is delimited by the two symmetric unstable fixed points *φ*_{1,2}=±*α*. The overturned positions *φ*=±*π*/2, not reported in figure 2, represent the practical failure of the system. The thick lines in figure 2 are the heteroclinic loop connecting the two saddles *φ*_{1} and *φ*_{2} and representing the significant part of their coincident stable and unstable manifolds. They split under the action of damping and excitation.

Despite its simplicity, the Housner model has a very intricate and complex behaviour, with characteristic features of applied nonlinear dynamics (e.g. multi-stability, sensitivity to initial conditions, fractal basin boundaries, transient/steady chaos, etc.) and specific aspects of non-smooth systems (e.g. grazing bifurcations, etc.). In this paper, we focus attention on the overturning of the rest position, which deserves a technical interest in several applications, and has been studied previously (Ishiyama 1982; Fielder *et al*. 1997; Lenci & Rega 2003*a*).

To illustrate the question, we report in the (*ω*,*γ*_{1}) parameter space of figure 3 the overturning chart of the rest position in the case of harmonic excitation (*N*=1 in equation (2.2)), which is commonly used in the literature dealing with periodic excitations, and is therefore considered as a reference to measure the developments proposed in this work. Figure 3 is obtained by integrating the equation of motion for given values of *ω*, *γ*_{1} and *ψ*_{1}, starting from the rest position up to 30*T*, and drawing a white (grey) point, if the block is toppled (untoppled). The position 30*T* is assumed as a measure of the length of the transient within which overturn occurs, if any (Fielder *et al*. 1997). This choice seems to balance the opposite requirements of accuracy and time-saving of the analysis, and is supported by the fact that, in the case of an earthquake, the excitation—though not periodic—has only a finite number of oscillations, comparable with the considered transient length.

Three different regions are identified in figure 3. For low excitation amplitudes, the block does not overturn at all, while for large amplitudes it always overturns. The intermediate region is a strip of fractal behaviour where toppling is highly sensitive to small parameter variations. It is bounded from below and above by the curves and , which can be determined analytically.

The lower curve corresponds to the *heteroclinic bifurcation* threshold, and its analytical expression is (see Lenci & Rega (2005) and §3*a*)(2.3)This is a very important dynamical event, whose occurrence starts the erosion of the safe (in-well) basins of attraction, which finally leads, after an involved series of dynamical phenomena, to overturning of the rest position and complete destruction of the safe basins. Thus, the heteroclinic bifurcation triggers the overturning, although its actual occurrence is also related to other subsequent topological events. To illustrate these facts, we have reported in figure 4*a* the basins of attraction for an excitation amplitude just below . Accordingly, the white-greys (i.e. in-well versus out-well) boundary is regular, although there are four safe basins, corresponding to four coexisting periodic attractors (Hogan 1989), with in-well fractality owing to the homo/heteroclinic intersection of other, non-hilltop, saddles, which are not of interest for the scope of the present work.

The erosion of the safe basin above is illustrated in figure 4*b*, which clearly shows the penetration of fractal white tongues from the out-of-well solution within the grey basins. Touching of the white with the point (0, 0) corresponds to actual overturning of the rest position, while complete disappearance of the grey basins marks the end of confined dynamics.

Besides its theoretical interest, the heteroclinic bifurcation threshold represents a lower *practical* approximation of the actual curve above which overturning certainly occurs for some given excitation phase (Lenci & Rega 2003*a*). In fact, above , the distance between toppling (white) basin and rest position decreases more or less quickly, so that the incidental overturning due to imponderable events is more and more likely. Thus, below , the rest position is safely protected from toppling by the stable manifolds of the hilltop saddles, whereas above it there is dangerous impending overturning.

On the other hand, the upper curve corresponds to the *immediate* *overturning* threshold, i.e. the excitation amplitude above which there exists an excitation phase *ψ*_{1} such that the rest position topples down without transient oscillations in the potential well. Its analytical expression is (see Lenci & Rega (2003*a*) and §3*b*)(2.4)

An example is reported in figure 5*a*, which shows how the solution leaves the rest position, oscillates around the right hilltop saddle and then escapes without impacts, i.e. without going back to the line *φ*=0. On the other hand, below , the solution may not overturn at all (figure 5*b*) or may overturn after transient in-well oscillations (figure 5*c*), this alternative being very sensitive to system parameter values, as confirmed by the closeness of *γ*_{1} in figure 5*b* and *c*.

The present work is devoted to investigate how the previous scenario, which refers to harmonic excitation, is modified in the case of *generic periodic* excitations. For the sake of clarity, and since they lead to different mathematical treatments, in the following we distinguish and compare the cases in which either of the two curves and increase (§3) or decrease (§4) by varying the shape of the excitation. Working with and makes sense because, as shown: (i) they bound the fractal strip of uncertain behaviour, (ii) they represent the boundary of immediate overturning and a reliable approximation of the threshold of safe region (figure 3), and (iii) they can be detected analytically, thus permitting closed form developments.

The unharmonic excitations are chosen shrewdly. In fact, the second key point of this paper is the search for some kind of ‘optimal’ (to be rigorously defined in the following) excitations permitting most advantageous increments or decrements of and . The first case is interesting when one is able to modify the shape of the excitation with the aim of increasing the amplitude threshold for the critical event. Here, the point of view is that of an operator which can correct a given external excitation by adding appropriate superharmonics, which do not modify the overall period. Accordingly, this case is called *control*, and will be investigated in §3.

In contrast, the second case has interest when one wants to identify the excitation shape producing maximal decrease in the amplitude threshold for the critical event. Here, the point of view is that of a designer dealing with environmental and partially unknown excitations, and wishing to know, for a safe design, the excitation producing the worst effects on the system. This case is the opposite of the previous one, and hence is called *anti-control*. It will be investigated in §4.

## 3. Control

We look for excitations permitting one to increase the heteroclinic bifurcation (§3*a*) and/or the immediate overturning (§3*b*) thresholds, namely, we are interested in controlling the nonlinear dynamics of the rigid block through increasing the amplitude thresholds triggering undesired events.

In this respect, it is useful to emphasize a few points. In fact, it is obvious that if we are able to modify (in particular to increase, see figure 3) the frequency and/or to reduce the excitation amplitude and/or to increase the damping, the dynamics are generally regularized and the overturning moved away. Analogously, it is easy to understand that if the addition of the controlling superharmonics is detrimental to the amplitude of the principal harmonic (such that, e.g. a measure of the excitation power like remains constant), this once again trivially leads to safer dynamics, as high-frequency harmonics entail high critical thresholds (see again figure 3). Thus, the non-trivial control problem consists in adding control superharmonics to a given, unchangeable, principal harmonic *γ*_{1}cos(*ωt*+*ψ*_{1}) with the aim of reducing its negative effects, i.e. with the aim of increasing and . In physical terms, the interesting problem is indeed controlling system dynamics by supplying energy, while controlling by removing energy or shifting its frequency content is conceptually easier. Accordingly, we rewrite equation (2.2) as(3.1)and assign *γ*_{1} the role of *overall excitation amplitude*, while the dimensionless parameters *γ*_{j}/*γ*_{1} measure the relative amplitude of the superharmonics corrections, i.e. they are the *shape* parameters to be cleverly chosen, together with the phases *ψ*_{j}, to optimize the performances.

### (a) Optimal control of the heteroclinic bifurcation threshold

This item has been studied in Lenci & Rega (2005), and only the main results useful to the present work are summarized herein.

On the Poincaré sections , the distance between the (upper/lower) stable and unstable manifolds of the heteroclinic loop (see figure 2) can be computed exactly, thanks to the piece-wise linearity of the system,(3.2)where , *τ* is a phase parameter along the unperturbed loop and the quantities *d*_{0}, *d*_{1}(*ω*), *ϕ*(*jω*), and *β*(*jω*) are given in Lenci & Rega (2005). The condition *d*^{up,low}(*τ*)=0 (for some *τ*∈[0,T]) for heteroclinic intersections are given by(3.3)where represent the amplitude thresholds for the heteroclinic bifurcation of the upper (lower) manifolds. In the case of harmonic excitation, since *h*_{1}=1, , *M*^{up}=*M*^{low}=1, so that represents the bifurcation value in this reference case (compare with (2.3)). and depend on the *shape* of the excitation, i.e. on the added controlling superharmonics, only by means of the numbers *M*^{up} (*M*^{low}), and coincide only for ‘symmetric’ excitations satisfying . Otherwise, the unique curve of figure 3 splits in two curves related to the upper (lower) heteroclinic bifurcations.

The control acts increasing the critical thresholds and by varying the shape parameters *γ*_{j}/*γ*_{1} and *ψ*_{j}, *j*=2…*N*. To quantitatively measure the improvement obtainable with generic shapes with respect to the reference harmonic excitation, the *gains* are defined as follows:(3.4)

The goal is to increase the gains as much as possible, and two different strategies can be pursued. Indeed, we can control either *only* the upper (lower) heteroclinic bifurcation, irrespective of what happens to the other, or *simultaneously* the upper and lower heteroclinic bifurcations. The first approach (*one-side* control) is aimed at obtaining a topologically ‘localized’ control, whereas the second one (*global* control) is aimed at controlling, on an average, the ‘whole’ phase space. In other words, one-side control prevents only from one of the two possible topplings (towards right or left), while global control prevents from both.

#### (i) One-side control

Let us consider *only* the upper (respectively lower) heteroclinic bifurcation. In this case, the demand of increasing the heteroclinic bifurcation as much as possible by adding controlling superharmonics yields the following optimization problem:(3.5)

#### (ii) Global control

If we want to increase *G*^{up} and *G*^{low} *simultaneously*, we must increase their minimum value, namely, we must solve the optimization problem:(3.6)As expected, the solution of (3.6) is seen to satisfy the condition *G*^{up}=*G*^{low}=*G*, i.e. . Accordingly, the problem can be rewritten in the simplified form(3.7)where due to the constraint, there is no ambiguity in the definition of the gain, and the test functions are restricted to belong to the class of ‘symmetric’ excitations.

The optimization problem (3.7) is just the problem (3.5) with the addition of the constraint accounting for simultaneous control of the two heteroclinic bifurcations. Accordingly, the optimal gain in the case of global control is expected to be lower than that corresponding to one-side control. This enlightens the differences between the two strategies, which are complementary rather than competing with each other. In fact, one-side control provides ‘high’ optimal gain (we can theoretically double the critical threshold, see table 1), which is, however, restricted to the upper (lower) part of the phase space, whereas global control gives ‘low’ optimal gain (not larger than 27.32%, see table 1) but eliminates both heteroclinic bifurcations.

The solutions to problems (3.5) and (3.7) are attained for in-phase superharmonics of the function *h*(*m*), i.e. . The Fourier coefficients *h*_{j} for increasing the number of superharmonics are given in Lenci & Rega (2004). Optimal global solutions have no even superharmonics (*h*_{2j}=0). Thus, the optimization problem (3.7) picks the solution in the class of functions , for which the constraint is trivially satisfied being −*h*(*m*+*π*)=*h*(*m*).

Theoretical gains correspond to an infinite number of controlling superharmonics, and represent upper bounds of those obtainable when considering optimal solutions with a finite number of superharmonics, as mandatory in practical implementations. Reduced gains are increasing functions of the number of superharmonics, and excellent results can actually be obtained even by using few of them (see table 1). The one-side control gains are always considerably larger than the global ones. The optimal excitations are given by the expression(3.8)

### (b) Optimal control of the immediate overturning threshold

The immediate overturning threshold is defined as the minimum excitation amplitude above which there exists an excitation phase such that the rest position topples down without transient oscillations in the potential well (see figure 5*a*). In an equivalent manner, it can be defined as follows, this way better emphasizing the fundamental role played by the excitation phase.

If, for a given value of the excitation frequency, we slowly increase the excitation amplitude, the value corresponding to first overturning without impacts on the base depends *strongly* on the assumed excitation phase. The minimum value when the phase ranges in the interval [0,*T*] is defined as the immediate overturning threshold (for the fixed *ω*). This definition is motivated by the fact that, in practical applications, the excitation phase is often unknown, and one wants to protect the system from the worst one which, in general, is also a function of *ω*. Accordingly, in this section we consider the excitation(3.9)which is just a time shift of (3.1). *f* is the overall ‘phase’ explicitly introduced to stress the strong phase-dependence of the problem, which holds even in the case of harmonic excitation (Lenci & Rega 2003*a*).

Starting from the general solutions *φ*(*t*) of equations (2.1*a*,*b*) under the action of (3.9), in Lenci & Rega (2003*a*) the following relations have been obtained for the amplitudes *B*_{1,2} of the exponentially diverging term which governs the short-term behaviour of the two rocking solutions, toward right or left, ensuing from the rest position *φ*(0)=0, ,(3.10)with *λ* and *μ* being given in (2.3), , and given quantities , , *ϕ*(*jω*) and *Χ*(*jω*). The condition *B*_{1}>0 (respectively *B*_{2}<0) has been proposed in Lenci & Rega (2003*a*) as an approximate criterion for immediate overturning towards the right (respectively, towards the left).

If the overall excitation phase satisfying (respectively, ) is given, then the specific excitation amplitude for right (respectively, left) overturning without in-well oscillations reads:(3.11a)(3.11b)

On the contrary, if we look for the immediate overturning threshold, namely, we suppose that *f* is free or unknown then we get respectively:(3.12)

(3.13)

The curves are the amplitude thresholds for immediate overturning towards the right (left). In the case of harmonic excitation, since , , , so that represents the critical value in this reference case (compare with (2.4)). Note that and coincide only for ‘symmetric’ excitations satisfying , otherwise the unique curve of figure 3 splits in two distinct curves and .

We note the formal analogy between (3.12), (3.13) and (3.3), which suggests that the two problems of increasing and , though remaining conceptually distinct, are mathematically identical, the difference being only in the different definition of the functions *d*, *h*(*m*) and , . Thus, we can take advantage from the developments of the previous section, and define the gains(3.14)which measure the improvement obtainable with respect to the basic harmonic excitation.

The concepts of *one-side* and *global* controls remain unchanged, and the optimal excitations are still obtained by solving problems (3.5) and (3.7). The fact that the superharmonics of are in phase now means that , while the Fourier coefficients for increasing superharmonics number *N* are the same as in the case of control of homoclinic bifurcation. The worst phase *f* involved in the definition of the immediate overturning threshold is given by , where is reported in table 1. In turn, the optimal excitations can be written in the form(3.15)

### (c) Comparison of overall dynamics under the two control strategies

The optimal excitations obtained in §3*a*,*b* provide the same gains, so that we are able to increase the same ratio both and . Remarkably enough, this increment is frequency-independent, a property very useful in practical applications, which no longer holds for anti-control (§4). However, there are some differences between the two control strategies.

The optimal excitations (3.8) and (3.15) differ in the superharmonic amplitudes and phases, the former being perhaps more important for they are somehow related to the cost of control, while the latter are responsible for the correct synchronization of controlling terms.

To compare the two cases, we report in figure 6 the functions , where and are the superharmonic amplitudes of immediate overturning and heteroclinic bifurcation optimal controls, respectively. Note that is independent of *N*, because *h*_{j} disappears in the ratio. The picture shows that, even though the coefficients are similar for small and large values of *ω*, in the intermediate range is systematically smaller than *γ*_{j}, the difference being important even for low-order superharmonics. This means that it is easier to increase rather than .

This difference entails that, in general, we cannot control and simultaneously, the optimal control of one curve possibly leading to decreasing of the other, as shown in the following.

First, let us suppose that we are applying *global* optimal control of the heteroclinic bifurcation threshold (i.e. the excitation (3.8)), so that and , and we want to investigate the effects on the immediate overturning threshold. The corresponding function is, from equation (3.10),(3.16)and contains only even terms, so that . This implies that the symmetry is maintained, i.e. right and left immediate overturning occur for the same excitation amplitude, corresponding to the unique gain . Thus, *global optimal* control of heteroclinic bifurcation implies *global non-optimal* control of immediate overturning. However, contrary to *G*^{up}=*G*^{low}, which is constant and given in table 1, the gain of the uncontrolled curve now depends on *ω*, as depicted in figure 7*a*.

In the opposite case, let us suppose that we are using *global* optimal control of the immediate overturning (i.e. the excitation (3.15)), so that and , and we want to investigate the effects on the heteroclinic bifurcation. The corresponding function *h*(*m*) is, from equation (3.2),(3.17)where is a shift in the argument, which is not essential as it does not change the minima and the maxima. The symmetry is also maintained in this case, so that the non-controlled curve is subjected to the unique gain , which is again *ω*-dependent and is reported in figure 7*b*. Again, *global optimal* control of immediate overturning implies *global non-optimal* control of heteroclinic bifurcation.

Yet, what can be said as regards the extent of gain (or loss)? Figure 7*a* and *b* share some interesting qualitative properties. In both cases, *G*>1 only for small and large excitation frequency, while for medium values of *ω*, it is less than 1. Furthermore, the asymptotic limits of *G* for *ω*→0 and *ω*→∞ are exactly the gains of the controlled curve. Thus, we can distinguish three qualitatively different regions of behaviour, labelled as a, b and c in figure 7*a* and *b*. For very small and very large values of *ω* (regions a), we are actually able to *optimally* control both and simultaneously by acting on only one of them. For small and large values of *ω* (regions b), the optimal control of, say, implies also sub-optimal control of . Instead, in the central region c, the control of one curve implies a decreasing of the other, i.e. a worsening of the behaviour of the uncontrolled curve, the larger the increment of the former, the larger the decrement of the latter.

However, there are also differences in the shape of the two sets of curves, and in the extent of regions a, b and c in the two cases. More importantly, in the central region the decrement of due to control of (figure 7*a*) is very high, whereas the decrement of due to control of (figure 7*b*) is minor (note the different scale of the two figures), though not negligible. Roughly speaking, this means that the effects on system dynamics of controlling are strong, while those of controlling are mild. This agrees with the findings in figure 6.

The decrease of the uncontrolled curve in the central region c has opposite effects in terms of system response. In fact, the increase of accompanied by a considerable decrease of implies that under this non-harmonic excitation the medium fractal strip of figure 3 of impending overturning behaviour is strongly reduced. This theoretical prediction is confirmed by the numerical results in figure 8*a*, which reports the overturning chart and the curves and under global optimal control (*N*=3) of the heteroclinic bifurcation. For the sake of comparison, it also reports the corresponding curves with harmonic excitation (which are exactly those of figure 3). Note that is constantly increased by 15.47%, while, consistent with figure 7*a*, is initially increased and then decreases becoming approximately 70% of its reference curve.

On the contrary, the increase of with slight decrease of implies that the fractal strip somehow enlarges, as numerically confirmed by figure 8*b*, which reports the overturning chart and the curves and under global optimal control (*N*=3) of the immediate overturning. Now, is constantly increased by 15.47%, while initially increases (this is not visible in figure 8*b* due to graphical approximations) and then, consistent with figure 7*b*, it slightly decreases becoming approximately 95% of its reference curve.

Previous comparisons refer to global control. Similar results can be obtained by comparing one-side controls, which, however, are less interesting. In fact, if we control only one heteroclinic bifurcation, the penetration of unsafe tongues in the safe basin of attraction occurs from the other side, and the triggering of overturning is not eliminated (indeed, isolated tests show that it is enhanced!). Analogously, elimination of immediate overturning on one side is not satisfactory if not accompanied by simultaneous elimination of overturning on the other side.

## 4. Anti-control

In this section, we investigate the problem opposite to that considered in §3, and look for excitations permitting one to decrease, as much as possible, the heteroclinic bifurcation (§4*a*) and the immediate overturning (§4*b*) thresholds. In fact, contrary to control, anti-control consists in enhancing the undesired event, usually with the purpose of using it and sometimes, as in this work, with the indirect purpose of preventing it.

Here, as in §3, *δ*, *r* and especially *ω* are kept fixed, otherwise the problem is trivial. However, the change in point of view implies an important change in the approach to the problem. In fact, in the present case there is no longer a given harmonic to be controlled by added superharmonics, but rather a ‘completely’ *free* periodic excitation. Accordingly, if we continue to use *γ*_{1} as the excitation amplitude, it would be easy to reduce and to zero (e.g. by choosing *γ*_{2}/*γ*_{1} large enough), and the problem would become meaningless. Thus, we need an overall amplitude measure, which takes into account the single amplitudes of all superharmonics, and we must look for the *worst* excitation in the class of shapes sharing the same magnitude, even accepting that the amplitude *γ*_{1} of the first harmonic reduces, which has no longer a special meaning.

Physical considerations support previous arguments, since it is not difficult to trigger undesired dynamical events, in particular overturning, by increasing the input energy. The interesting question is instead that of decreasing and by keeping fixed, if not even reducing, the input energy. We use(4.1)as the overall amplitude. This choice is also motivated by the fact that *P* represents a measure of the average power supplied to the system by the ground oscillations, and is involved in earthquake engineering (Uang & Bertero 1990). It is worth to note that for *N*=1, it is so that the amplitude measures for control and anti-control basically coincide for harmonic excitation. This occurs *only* for this excitation, and permits easy re-interpretation of the results of figure 3 in terms of *P*.

### (a) Optimal anti-control of the heteroclinic bifurcation threshold

For the purposes of this subsection, it is useful to rewrite the distance (3.2) between stable and unstable manifolds in the equivalent form:(4.2)The upper (respectively, lower) heteroclinic intersection can occur if and only if *d*^{up}(*τ*) (respectively, *d*^{low}(*τ*)) has simple zeros for some values of *τ*∈[0,T]. This leads to the associated conditions for intersection(4.3)(4.4)which are equivalent to (3.3). Contrary to §3*a*, here the aim is promoting heteroclinic intersections, i.e. looking for excitation shapes which *increase* the number *m*^{up,low} (whereas in control we were involved in *decreasing* them). Again, this can be pursued increasing only *m*^{up} (respectively, *m*^{low}) irrespective of what happens to the other (*one-side* anti-control), or we can simultaneously increase *m*^{up} and *m*^{low} (*global* anti-control).

#### (i) One-side anti-control

In this case, the demand of promoting *only* the upper (respectively, lower) heteroclinic intersection as much as possible by varying the shape of the excitation yields the following optimization problem:(4.5)where the last constraint is added because, as previously noted, the problem is not trivial, only if we fix the input energy. If we first maximize with respect to *ψ*_{j}, it is possible to show that problem (4.5) reduces to(4.6)and can be solved by means of the Lagrange multipliers theorem by looking for the (unconstrained) minimum of the augmented function(4.7)

The anti-control optimal excitation turns out to be(4.8)while *γ*^{low}(*t*) is obtained by substituting *π* with 0 in (4.8), i.e. *γ*^{low}(*t*)=−*γ*^{up}(*t*), and the condition for heteroclinic intersections of upper (respectively, lower) manifolds reads(4.9)where *P*_{cr} represents the amplitude threshold for heteroclinic bifurcation of the optimal one-side anti-controlled excitation (4.8).

As in the case of control, it is interesting to compare with the heteroclinic bifurcation thresholds of anti-controlled and (reference) harmonic excitations by means of the *optimal* gain (which should be likely named ‘anti-gain’)(4.10)where the fact that *γ*_{1}=√*P* of harmonic oscillations is made use of. The gains for optimal anti-control of upper heteroclinic bifurcation are depicted by full lines in figure 9 for various values of *N*.

However, in the case of one-side control, the lower and upper heteroclinic bifurcations no longer occur simultaneously, because the symmetry of the excitation is lost, so that the effects of the optimal anti-control of, say, the upper heteroclinic bifurcation on the *non*-anti-controlled lower heteroclinic bifurcation must be investigated. Thus, let us suppose to use the excitation *γ*^{up}(*t*) given by (4.8). From (4.4) we have(4.11)and from the condition *m*^{low}>*d*_{0} we get(4.12)The ratio(4.13)is the gain of the *non*-anti-controlled heteroclinic bifurcation to be compared with the optimal gain of the anti-controlled heteroclinic bifurcation, given by (4.10). It is depicted in figure 9*a* by dotted lines for various values of *N*.

The main property emphasized by figure 9*a* is that, contrary to what happens in the case of control, the *optimal* anti-control gains are *ω*-dependent. In particular, for each given number of superharmonics *N*, there are basically three different regions. For low excitation frequency, *G*s tend to 1/√*N*, for medium frequency they are more or less independent of *ω* and *N*, and approximately equal to 0.95, while for large frequency they tend to (which approaches √6/*π*=0.78 when *N*→∞). This means that the differences with harmonic excitations are very small for medium values of *ω*, little for large values of *ω* and large for small values of *ω*. Thus, if we use harmonic excitation as representative of excitations of given frequency and energy, but unknown shape, the analysis will be reliable for medium *ω* values, adequate for large *ω* values and inaccurate for small *ω* values.

Figure 9*a* also shows the characteristic behaviour of the gain of the *non*-anti-controlled curve. First, it is always larger than 1. This means that, when we optimally decrease the threshold of, say, the upper heteroclinic bifurcation by varying the shape of the excitation, we automatically increase the threshold of the lower heteroclinic bifurcation. In other words, one-side anti-control is effective only on the anti-controlled part of the phase space, where the ‘worst’ excitation concentrates all efforts, and ineffective elsewhere. This fact agrees with similar results obtained in the case of one-side control.

Another property of *G*^{non-anti-controlled} emphasized by figure 9*a* is a kind of resonance observed for certain (relatively small) values of the excitation frequency. This implies that in this range, the behaviour of the system has been strongly asymmetrized, while elsewhere there is only a moderate asymmetrization.

Finally, we note that while for one-side control there is a theoretical constant upper bound on the gain, for one-side anti-control there is a frequency-dependent lower bound, which tends to 0 for *N*→∞ and *ω*→0.

#### (ii) Global anti-control

If we want to increase *m*^{up} and *m*^{low} *simultaneously*, we must increase their minimum value, namely, we must solve the optimization problem:(4.14)

Similarly to what happens in the case of control, (i) problem (4.14) is a constrained version of (4.5), so that global anti-control is still theoretically less performant, i.e. with a gain closer to 1, and (ii) the optimal solution has only odd terms, thus automatically satisfying the optimality condition *m*^{up}=*m*^{low}: it is still given by (4.8), with no even terms in the summation. Analogously, the gain is given by (4.10) by taking into account only odd terms, and is depicted in figure 9*b* for the various values of *N*. Comparison of figure 9*a* and *b* confirms that *G*^{one-side}≤*G*^{global}, namely, one-side anti-control is more effective as it permits major decrements of the heteroclinic bifurcation threshold. Figure 9*b* shares the same qualitative properties of figure 9*a*, with only quantitative differences. For example, in the medium range of *ω* the gain (*ω*- and *N*-independent) is approximately equal to 0.99, thus confirming, at least in this range, the reliability of analyses based on the harmonic excitation.

### (b) Optimal anti-control of the immediate overturning threshold

To investigate the anti-control of immediate overturning, which is substantially similar, though conceptually distinct, to that of heteroclinic bifurcation, we follow the same—though opposite—guidelines as those in §3*b*, which is referred to for the main features of the subject. The starting point is to re-write (3.11a) in the equivalent form(4.15)Then, looking for the right (respectively, left) immediate overturning threshold, i.e. supposing *f* is free or unknown, entails(4.16)(4.17)which are equivalent to (3.12) and (3.13), respectively.

Promoting immediate overturning yields increasing the numbers (instead of decreasing them, as in control), and this can be done following a *one-side* or a *global* anti-control approach, as in the previous section. By a correct, and easy, identification of different symbols, we note that (4.16) and (4.17) are formally identical to (4.4) and (4.3), respectively. Furthermore, we are still aimed at increasing as much as possible. Therefore, we can directly use the results of the previous section, and conclude that the one-side optimal excitation for worst immediate overturning toward the left is(4.18)while *γ*^{r}(*t*)=−*γ*^{l}(*t*) (note that the worst phase *f* is equal to 0). The left (respectively, right) immediate overturning thresholds become(4.19)while the *optimal* anti-control gains are(4.20)Those for left immediate overturning are depicted by full lines in figure 10*a*, which also reports by dotted lines the gains of *non*-anti-controlled immediate overturning (right in this particular instance), given by(4.21)

The global control is simply obtained by considering only the odd terms in (4.18), (4.19) and (4.20). The gains are reported in figure 10*b* for the sake of comparison with figure 10*a*. Obviously, the thresholds of right and left immediate overturning are now coincident.

The asymptotic limits for *ω*→0 and *ω*→∞ of the gains in figures 10*a* and *b* are exactly those of the corresponding figures 9*a* and *b*, the single qualitative difference being that the former are monotonic increasing functions of the excitation frequency. Thus, in the case of anti-control of immediate overturning, we basically have two (instead of three) regions, one for small excitation frequency, where harmonic excitation analysis is inadequate, and another for medium and large values of *ω*, where considering a generic excitation shape yields only minor differences.

### (c) Comparison of overall dynamics under the two anti-control strategies

A more detailed comparison between anti-controls of heteroclinic bifurcation and immediate overturning is performed in this section. To begin with, we investigate the differences between optimal excitations (4.8) and (4.18), by considering the ratios(4.22)between the amplitudes of each term, which, contrary to what happens in the case of control (see figure 6), slightly depend on *N*, as shown in figure 11 for two relevant sample values. Another difference is that, apart from the case *j*=1, the numbers are almost everywhere larger than 1, namely, the anti-control of immediate overturning requires, with respect to anti-control of heteroclinic bifurcation, less power in the first harmonic and considerably more power in the higher harmonics, thus, it is harder to decrease rather than , contrary to control. However, as in the latter case, the two anti-controls coincide for low- and high-excitation frequencies, and this proves that in these ranges, optimal anti-control of one curve entails optimal anti-control of the other, a property that, still as in the case of control, no longer holds for medium frequencies.

From (4.10), (4.20) and (4.22), note that(4.23)a relation which, along with figure 11, shows that *G*^{imm.over}<*G*^{heter.bif.}, namely, the anti-control of immediate overturning is always theoretically more performant.

Then, we investigate the effects of the anti-control of one curve on the other, non-anti-controlled, curves, namely, an issue corresponding to those in figure 7*a*,*b* for *global* control which, however, is herein more involved for we are dealing with *one-side* anti-control. To begin with, suppose that we are using optimal anti-control of the upper heteroclinic bifurcation (i.e. the excitation (4.8)), so that and , and we want to investigate the effects on the immediate overturning threshold. From (4.16) and (4.17) we have that(4.24)and(4.25)where the right-hand sides are the amplitude thresholds for the considered excitation, which split from the unique , the excitation being no longer symmetric. The gains are then(4.26)and are depicted in figure 12*a*.

Figure 12*a* shares the same qualitative shape as figure 9*a*, to which it should be compared, because both correspond to the same excitation *γ*^{up}(*t*). It shows that for small and large excitation frequency, optimal one-side anti-control of heteroclinic bifurcation entails optimal one-side anti-control of immediate overturning, and this is still obtained by a worsening of the non-anti-controlled part of the phase space. In the medium range of frequency, on the other hand, we have an opposite situation, and while anti-control of heteroclinic bifurcation is effective in reducing its own critical threshold (though not to a large extent, see figure 9*a*), it does not change or even slightly increases the threshold for immediate overturning, thus confirming the different behaviour of the two dynamical events for intermediate values of *ω*, also observed in the case of control (figure 7).

On the contrary, let us suppose that the excitation is *γ*^{l}(*t*), namely, equation (4.18), and we want to study the effects on the heteroclinic bifurcation threshold. We have and , so that from (4.3) and (4.4) we get(4.27)and(4.28)where the right-hand sides are the amplitude thresholds for the considered excitation, which split from the unique , the excitation being no longer symmetric. The gains are then(4.29)and are depicted in figure 12*b*.

Figure 12*b* must be compared with figure 10*a* because both correspond to the same excitation *γ*^{l}(*t*). Firstly, we note that they do not have the same shape, a fact that implies a different qualitative (besides quantitative) effect of one-side anti-control of immediate overturning on the heteroclinic bifurcations. However, comparable gains still occur for *ω*→0 and *ω*→∞, being in both cases accompanied by a worsening in the non-anti-controlled part of the phase space. Instead, in the central part, the effectiveness of one-side anti-control of immediate overturning is accompanied by an increase (which to the aim of the present section means ineffectiveness) of both heteroclinic bifurcation thresholds.

Previous theoretical predictions are compared with numerical simulations in figure 13, which reports the overturning charts obtained with the simplest (*N*=2) non-harmonic optimal anti-control excitations *γ*^{up}(*t*) (figure 13*a*) and *γ*^{l}(*t*) (figure 13*b*), respectively. Splitting of both left/right immediate overturning thresholds and upper/lower heteroclinic bifurcations thresholds is clearly visible. However, the latter is always modest, according to closeness to 1 of the related gains (see figures 9a and 12*b*), while the former varies. It is modest in figure 13*a* (consistent with figure 12*a*) and more pronounced in figure 13*b* (consistent with figure 10*a*). Thus, the triggering of safe basin erosion is practically unchanged with respect to the harmonic excitation, whereas immediate overturning is basically not modified under anti-control of heteroclinic bifurcations, and reduced to a certain extent when being directly anti-controlled.

This has different consequences on the fractal strip of intermediate behaviour. In fact, the lower bound practically does not change, whereas the upper bound does not change (changes) in figure 13*a* (13*b*). Accordingly, the fractal strip of figure 3 (harmonic excitation) is reduced under one-side anti-control of immediate overturning (compare with control, figure 8*b*) and remains basically unchanged under one-side anti-control of heteroclinic bifurcation (compare with control, figure 8*a*).

It is worth noting that, if considering higher order optimal anti-controls, i.e. if increasing the number of superharmonics, good agreement between theoretical predictions and numerical simulations is still observed.

In any case, the system experiences immediate overturning for the lowest of the two different theoretical curves and . As this minimum is lesser than the coinciding ones in the case of global anti-control, and as herein the aim is triggering the dynamical event, i.e. reducing the critical thresholds, it is clear that in the case of anti-control, the most appropriate excitation is the one-side optimal, contrary to control where it is the global optimal (see the discussion at the end of §3*c*). This point can be made for anti-control of heteroclinic bifurcations, too, and explains why the comparison of the effects of global anti-control is not reported.

## 5. Conclusions

A systematic theoretical investigation of control/anti-control of the nonlinear dynamics of a rocking block has been made through the analysis of two curves, the *heteroclinic bifurcation* and the *immediate overturning* thresholds, characterizing the system response in excitation parameters space in terms of overturning behaviour. The thresholds have been detected analytically, and their modifications due to the generic *shape* of the periodic excitation have been investigated in depth.

The two opposite cases of increase (*control*) and decrease (*anti-control*) of these curves have been considered separately; however, looking in both cases for the excitation permitting the maximum variation of the considered critical threshold. The differences between control and anti-control are mostly of a technical nature, and basically consist in the different definition of the ‘amplitude’ of the excitation and in the different nature of the optimization problems. These have been solved in closed form, by determining also the optimal excitations with an increasing number of superharmonics.

The concepts of *one-side* or *global* control, previously introduced by the authors, are seen to apply to the present investigation. They are based on the fact that the critical events are indeed double, and coincide only for symmetric excitations. Roughly, global control looks for the optimal excitation being still symmetric, while one-side control ignores this point and permits a major increase/decrease of the critical curves. The former approach is shown to be more appropriate in the case of control, the latter in the case of anti-control, where it also entails splitting of the reference threshold.

A detailed analysis of the effects of control (anti-control) of one curve on the other, uncontrolled (non-anti-controlled), curve has been performed. For small and large excitation frequencies, control (anti-control) of one curve automatically entails control (anti-control) of the other. The situation is more involved in the range of medium frequencies, where in general increase (decrease) of one curve implies decrease (increase) of the other.

*Theoretical* predictions have been compared with *numerical* simulations by means of overturning charts, which show good agreement between theoretical and numerical aspects. They permit further understanding of the practical effects of optimal excitations.

The analysis has been developed on a piece-wise linear model of rigid block, which is accurate only for slender blocks. However, the same ideas can also be applied to a fully nonlinear model, possibly by means of Melnikov-like perturbative analyses for detecting the relevant curves. More generally, the proposed control and anti-control techniques can be applied to different mechanical systems with the associated thresholds of interest. While this has been done for control (Lenci & Rega 2004), it should be pursued also for anti-control to show that, indeed, the techniques are general and not specific to the rigid block. This is left for future works.

As a meaningful by-product, the present analysis conducted with non-harmonic excitations shows that only in special cases are the differences in the dynamical response of an archetype mechanical oscillator with respect to harmonic excitation negligible. Accordingly, all of the results obtained with the latter in the literature, under the tacit assumption that it is a good representative of generic periodic excitations, must be considered with care.

## Footnotes

One contribution of 15 to a Theme Issue ‘Exploiting chaotic properties of dynamical systems for their control’.

- © 2006 The Royal Society