Bifurcations of the Controlled Escape Equation Tobias Gaer Institut für Mathematik, Universität Augsburg 86135 Augsburg, German gaer@math.uni-augsburg.de Abstract In this paper we present numerical methods for the analsis of nonlinear autonomous control sstems and two conditions, local accessibilit and an inner-pair condition, under which the can be applied. These methods can be etended to work also for sstems with time-periodic right hand side. In particular, the escape equation with sinusoidal driving term and additional control is analed. We will show that its stabilit behavior undergoes interesting bifurcations when the range of the control influence is varied. 1 Introduction In this paper we consider a famil of nonlinear control sstems ẋ(t) = f((t), u(t)) (1.1) on a connected smooth manifold M with dimension d where f : M R m TM is smooth and the controls u are taken from the set U ρ := {u L (R, R m ), u(t) ρ U for almost all t R} for a non-void, conve, and compact subset U of R m with intu and a parameter ρ [, ρ ]. Relevance of the parameter dependence will be indicated b a superfi ρ. We assume for ever initial value M and ever u U ρ that there eists a unique solution φ(t;, u), t R, with φ(;, u) =. Furthermore, b restricting ourselves to a compact invariant subset of the state space we can focus on compact M onl. Throughout we assume the sstem to be locall accessible for all ρ >, i.e. for all M and all T > one has into ρ,+ T (), and intoρ, T (), where Oρ,+ T () = { M, = φ(t;, u) with t T and u U ρ } and O ρ, T () = { M, = φ(t;, u) with t T and u U ρ } are the positive and negative reachable sets from. The limit behavior of this sstem is determined b its control sets, i.e. maimal subsets of M where complete approimate controllabilit holds, and its chain control sets (see [1]). Under an inner-pair condition the chain control sets and the control sets genericall coincide. We will see in section 2 that this inner-pair condition holds for n-th order equations with additive control and therefore under change of the parameter ρ the control sets will change continuousl for all but countabl man ρ [, ρ ]. If the right hand side of the control sstem has a t-periodic driving term as onl eplicit time dependence, the time can be interpreted as a further space dimension and control sets and chain control sets can be studied for the corresponding sstem without direct time dependence. We show that if in this case control 1
sets merge under variation of the parameter ρ, then the merge over the whole time-period at once. Onl in ver simple cases can control sets be found analticall. Thus numerical methods are an important tool for the analsis of control sstems. In section 3 we briefl present a method for the computation of control sets that was developed b Solnoki (see [7]) and is based on an algorithm for the computation of relative global attractors and unstable manifolds of dnamical sstems b Dellnit, Hohmann and Junge (see [3] and [4]). In section 4 finall we will clarif our findings b numerical results for the controlled escape equation. 2 Control sets and chain control sets Control sets are maimal subsets of the state space where complete approimate controllabilit holds. A control set C is called invariant if clc = cl{φ(t;, u), u U, t }. If we allow the solutions to make small jumps, we are led to the notion of chain-controllabilit: For ɛ, T >, and, M a controlled (ɛ, T )-chain from to is given b n N,, 1,..., n M, u,..., u n 1 U, t,... t n 1 T with =, n =, and d(φ(t j ; j, u j ), j+1 ) ɛ for all j =,..., n 1. A chain control set E M is a maimal subset such that (i) for all, E and all ɛ, T > there is a controlled (ɛ, T )-chain from to, and (ii) for all E there is u U such that φ(t;, u) E for all t R. Ever control set is contained in a chain control set. The following inner-pair condition guarantees much closer a relationship between control sets and chain control sets (cf. [1]): Definition 2.1. If for all ρ 1, ρ 2 [, ρ ], ρ 1 < ρ 2, and for all M, u U ρ 1, there is T > such that φ(t ;, u) into +,ρ 2 (), then the famil of control sstems (1.1) is said to fulfill the inner-pair condition. Theorem 2.1. Let E ρ 1 M be a chain control set of (1.1) ρ 1, and for each ρ [ρ 1, ρ ] let E ρ M be the unique chain control set of (1.1) ρ with E ρ 1 E ρ. Moreover let the famil (1.1) be locall accessible and fulfill the inner-pair condition. Then for each ρ there is precisel one control set D ρ with E ρ 1 intd ρ, and for all but countabl man values of ρ the following holds: (i) cld ρ = E ρ. (ii) The mapping ρ cld ρ is continuous with respect to the Hausdorff metric. At the discontinuities control sets merge and possibl change their stabilit behavior. The inner-pair condition can be etended to time dependent sstems in the obvious wa and the following holds. Proposition 2.1. Consider the famil of n-th order sstems (n) (t) + f(t,, (1),..., (n 1) ) = b(t,,..., (n 1) ) u(t) (2.2) 2
where f, b : R n+1 R are continuous mappings and u U ρ. If there is α > such that b(t,, 1,..., n 1 ) α holds for all (t,, 1,..., n 1 ) R n+1, then (2.2) fulfills the inner-pair condition. If the right hand side in (1.1) is generalied to be eplicitl periodicall time dependent, the control setting can still be used. Let f : R M R m TM be smooth and a sstem be given b ẋ(t) = f(t, (t), u(t)). If f is T -periodic in the first coordinate: f(t,, u) = f(t + T,, u), we can consider the associated time-independent sstem ( ẋ ż ) = ( f(,, u) T/2 ) (2.3) where we identif = and = 2. If control sets for such a sstem have contact for one R/2Z, the have contact for the whole interval. Proposition 2.2. If D 1 and D 2 are control sets for the control sstem (2.3) and if there is (, ) R/2Z M such that (, ) cld 1 cld 2, then for ever R/2Z there is M such that (, ) cld 1 cld 2. Proof. For proofs of Propositions 2.1 and 2.2 see [5]. 3 Numerical approimation of control sets Control sets can not be approimated directl but the computation of chain control sets is almost as good according to Theorem 2.1. Solnoki developed three algorithms whose combination is a powerful tool for locating chain control sets (see [7] and [8]). The have in common that a compact subset Q of the state space is chosen and successivel divided in finer and finer partitions. A selection criterion in each step defines a decreasing sequence of subsets of Q. A graph algorithm finds strongl connected components of a directed graph associated to the discretied dnamics. In principle these strongl connected components converge to the chain control sets if the partition gets finer but memor and time consumption do not allow for fine partitioning in higher dimensions. The subdivision algorithm approimates viabilit kernels V (Q) relative to the start set Q. If Q contains precisel one chain control set E, then E = V (Q). If Q possibl contains more than one chain control set, then in some cases a continuation algorithm can be used to create a covering of precisel one chain control set, which allows the subdivision algorithm to be used. A combination of these methods together with smart selection of the set Q often produces good results. But for sstems that are sensitive to small changes in their initial conditions or for high dimensions resources are limited. In the special case of time-periodic sstems (2.3) one can spare one dimension b investigating Poincaré-cuts and using Proposition 2.2 to draw conclusions for the full dimensional sstem. 3
4 The controlled escape equation As an eample we consider the controlled escape equation ẍ(t) + γẋ(t) + (t) (t) 2 = F sin(ωt) + u(t) γ, ω, F >, t R. This equation has been intensivel studied for instance b Soliman and Thompson (see [6]). Interpreting the controls u as bounded noise terms it models the rocking movement of a ship on sea under the influence of periodic waves and some additional disturbance. According to Proposition 2.1 the inner-pair condition is satisfied. Transferring it into the form of sstem (1.1) leads to ẋ(t) ẏ(t) ż(t) = (t) γ (t) (t) (t) 2 ω + F sin((t)) + u(t) where M = R R R/2Z, u(t) U ρ := [ ρ, ρ], ρ ρ. We choose the parameter values ω =.85, γ =.1, F =.6. The sstem proves too complicated for a direct three dimensional investigation. Therefore we look at four Poincaré-cuts, at =, =.25 2, =.5 2, and =.75 2 and 1.5 investigate the discrete dnamics induced b the time-2-map. For the uncontrolled case ρ = we find four 1 fied points. At = FP1 is located close to (;.25) and is stable,.5 FP2 is near (.2;.4) and unstable, stable FP3 is close to (.3;.4) and unstable FP4 finall is close to (1; ). Choosing small neighborhoods of these fied points and using.5 the continuation algorithm we also find the unstable and stable manifolds of the unstable fied points 1 (see Figure 1). Now if we go to 1.5 1.5 1.5.5 1 1.5 =, ρ = Figure 1. For ρ = the picture shows two unstable (blue) and two stable (red) fied points. The unstable manifold of FP4 is depicted in green, its stable one in light brown. The unstable manifold of FP2 has the ellow color, its stable one is dark brown. 4, ρ =.5, according to Theorem 2.1 around these fied points develop invariant control sets D1 ρ and D3 ρ and variant control sets D2 ρ and D4 ρ for which we draw the domains of attractions as well. Finding a parameter value ρ for which there are three control sets is difficult. Onl for = /2 the graph algorithm is
able to resolve that at ρ =.85 the control sets D1 ρ and D2 ρ have merged whereas D3 ρ is still separated from them. But then Proposition 2.2 states that D3.85 must be separate for all and therefore we can cut out D1.5 and appl subdivision to the rest which gives the displaed pictures. At ρ =.1 the three control sets around the origin have merged to one invariant control set. At ρ =.13 the whole inner area has become transient since now the eit via the D4 control set is possible. Let A(D4 ρ ) denote the domain of attraction of D4 ρ and A(D2 ρ ) denote the domain of attraction of the variant control set that includes D2.5 (see Figure 2). We get the full 3D-pictures of the control sets b inserting the results at the selected four Poincaré-cuts, computing connecting orbits and appling subdivision afterwards (see Figure 3). 1..4.4.96 1 1.4 1..4.4.96 1 1.4.. 1. 1. 1.. 1. =, ρ =.5 1.. 1. =, ρ =.85 1..4.4.96 1 1.4 1... 1. 1. 1.. 1. =, ρ =.1 1.. 1. =, ρ =.13 Figure 2. For ρ =.5 control sets (red = invariant, blue = variant) have developed around the fied points. At ρ =.85 invariant D1 ρ has merged with variant D2 ρ forming D12.85 and lost its invariance. For ρ =.1 the sets D12.85 and D3.85 have merged and form one invariant control set in the middle. At ρ =.13 this control set in the middle has merged with the variant control set near (, 1) and formed one variant control set. Eit is now possible from everwhere. A(D4 ρ ) is depicted dark brown, A(D2 ρ ) is ellow. 5
1. 1. 2. 1. 2 1. 1. 1. /2. /2 1. 1. 1. 1.5 2 1. 2 1. /2 1. /2 1. 1. 1. References Figure 3 The 3D control sets for ρ =.5,.85,.1,.13. [1] F. Colonius, W. Kliemann, The Dnamics of Control, Birkhäuser, Boston, 2. [2] F. Colonius, W. Kliemann, Invariance under Bounded Time-Varing Perturbations, in: Proc. 11th IFAC International Workshop Control Applications of Optimiation, V. Zakhavov, ed., St. Petersburg (2), pp. 82-85. [3] M. Dellnit, A. Hohmann, A Subdivision Algorithm for the Computation of Unstable Manifolds and Global Attractors, Numer.Math. 75 (1997), pp. 293-317. [4] M. Dellnit, O. Junge, On the Approimation of Complicated Dnamical Behavior, SIAM J.Numer.Anal. 36 (1999), pp. 491-515. [5] T. Gaer, PhD-thesis, in preparation. [6] M.S. Soliman, J.M. Thompson, Integrit Measures Quantifing the Erosion of Smooth and Fractal Basins of Attraction, J.Sound and Vibration, 135 (3) (1989), pp. 453-475. [7] D. Solnoki, Algorithms for Reachabilit Problems, Shaker, Aachen, 21. [8] D. Solnoki, Computation of Control Sets using Subdivision and Continuation Techniques, in: Proc. 39th IEEE Conference on Decision and Control, Sdne (2), pp. 2135-2141. 6