Feedback stabilization of discrete-time homogeneous semi-linear systems

For discrete-time semi-linear systems satisfying an accessibility condition asymptotic null controllability is equivalent to exponential feedback sta-bilizability using a piecewise constant feedback. A constructive procedure that yields such a feedback is presented.


Introduction
We consider the problem of feedback stabilizability of homogeneous semi-linear discrete time systems, that is systems linear in the state where the entries of the transition matrix are functions of the control.This class is a generalization of the frequently studied bilinear systems.Systems of this form occur as linearizations with respect to the state only of nonlinear systems at singular points.Also they can be interpreted as systems in which the control a ects the parameter of a given system, see also 13].
In recent years there has been considerable progress in this area.For continuous time bilinear systems su cient conditions for feedback stabilizability have been presented by Ryan and Buckingham 19], Chen et al. 5], Celikovsk y 4] and Khapalov and Mohler 15].In the more general semi-linear case it has been recently shown in Gr une 10] that null controllability is equivalent to feedback stabilizability by discretized feedbacks and a numeric procedure for the calculation of stabilizing feedbacks has been presented.The analysis in this paper matrices which has been undertaken by Colonius and Kliemann,7], 8].Already in 9] the implications on stabilization of this approach have been studied.Furthermore the existence of a classical (measurable) feedback under the assumption of null-controllability is shown by Wang 22].For general nonlinear systems in continuous time it has been shown in 6] that asymptotic controllability is equivalent to feedback stabilizability by means of a sampled feedback.This approach, however, does not lead to exponential stabilization and is only constructive up to the fact that the knowledge of a control Lyapunov function is required.Su cient conditions for feedback stabilizability of bilinear systems in the discrete-time case are presented in Yang et al. 25] and Stepanenko and Yang 21].The methods employed in these references, however, use in a fundamental way that the system in non-homogeneous, i.e. that the origin is not a common xed point for all control values.General feedback stabilization schemes for discrete time systems have been presented by Simoes et. al. 3] and Lin and Byrnes 16], 17].The methods of the latter papers have been used to obtain smooth asymptotically stabilizing feedbacks for bilinear systems in 18] under the assumption that the uncontrolled system is Lyapunov stable.
In this paper we show how the discrete-time version of the results of Colonius and Kliemann which have been presented in Wirth 24] can be used to obtain discrete-time versions of the necessary and su cient conditions for feedback stabilizability.The proofs are constructive and we discuss numerical aspects of the constructed feedback.
The paper is organized as follows.In Section 2 we present the class of systems we consider and formulate the problem.Furthermore we introduce systems associated to the semi-linear systems, de ned on projective space.These systems are vital in the analysis of spectral properties of the original system and for this we review the relevant material from spectral theory of time-varying systems that is needed for the approach in this paper.The main theorem in Section 3 states that feedback stabilizability is characterized by a property of the Floquet spectrum.In the following Section 4 we construct stabilizing feedbacks using methods from optimal control theory to approximate optimal exponential growth rates along trajectories.Section 5 then shows how these results may be used in order to obtain a numerical scheme for the calculation of a piecewise constant exponentially stabilizing feedback.In the nal Section 6 we draw conclusions and comment brie y on the robustness properties of the proposed stabilization scheme.

Problem Formulation
We consider systems on R d of the form x(t + 1) = A(u(t))x(t) ; t 2 N ; (1) where A : Ũ !R d d is an analytic map, Ũ R m is open and connected, and the set of admissible control values satis es U Ũ. Let (t; u), t 2 N denote the evolution operator de ned by a sequence u 2 U N .We call system (1) asymptotically null-controllable if for every to singularity (t; u)x = 0 for some nite t may occur.System (1) is called (state) feedback stabilizable if there exists a map F : R d !U such that the system x(t + 1) = A(F(x(t)))x(t) ; t 2 N ; ( is globally asymptotically stable.If F can be chosen such that (2) is exponentially stable, then we call (1) exponentially (state) feedback stabilizable.It is the purpose of this paper to show that these concepts are equivalent if feedbacks can be chosen to be piecewise constant and to present a procedure for the calculation of exponentially stabilizing feedbacks.Note that it is inherent in this scheme that discontinuous feedbacks may occur.
Before presenting our general approach, let us brie y recall a special case of the results of 18].In this paper the authors consider systems of the form under the assumption that A is Lyapunov stable, i.e. we may choose P > 0 such that A T PA P 0. In the following P will always denote a matrix with these properties and furthermore kxk P := x T Px.Specializing to D = 0 we obtain a particular case of (1).In 18] a bounded globally asymptotically stabilizing feedback is shown to exist if for := fx 2 R d j (A s x) T (A T PA P)A s x = 0; s 0g S := fx 2 R d j (A s+1 x) T PB(A s x) = 0; s 0g we have \ S = f0g.Furthermore an explicit formula for the feedback is given, namely u(x) = (I + 1 2 B(x) T PB(x)) 1 B(x) T PAx (4) is globally asymptotically stabilizing but clearly in general not exponentially stabilizing.
However, exploiting homogeneity u can be modi ed to be exponentially stabilizing by choosing the feedback to be constant on rays f x ; x 2 R d n f0g; > 0g as can be seen from the following consequence of 18, Theorem 4].
Proposition 1 Consider a homogeneous bilinear system of the form (3) with D = 0 and assume that A is Lyapunov stable.If \ S = f0g then for any > 0 the system (3) is exponentially stabilized by the feedback de ned by F (0) = 0 and F (x) = holds.By the assumption \S = f0g it may be shown that for x(t; u) 6 = 0 there exists s 0 such that either u(x( ; u)) 6 = 0 for some 2 ft; : : :; t+s+1g or kx(t+s+1; u)k P < kx(t; u)k P .
Using continuity and compactness it follows that there exists a T 0 and a constant 1 > > 0 such that for any kx(0; u)k P = we have kx(T; u)k 2 P kx(0; u)k 2 P < kx(0; u)k 2 P .
By the homogeneity of the closed loop system de ned by F and the fact that F (x) = u(x) for kxk P = this implies kx(t + T; F )k 2 P kx(t; F )k 2 P kx(t; F )k 2 P (5) Hence exponential stabilization by the feedback F follows.
It is a further interesting fact that for the feedbacks F just de ned we have F (x) = F ( x).Thus F does de ne a smooth map on the projective space P d 1 , and in fact our general approach uses feedback maps induced by maps on P d 1 .As we will see these kind of maps su ce for the feedback stabilization of (1).The basic idea of the construction of the feedback is to obtain upper bounds on the exponential growth rates of the trajectories using ideas from optimal control.In a more general situation than the one considered in Proposition 1 however one cannot expect to obtain smooth or even continuous feedbacks using this approach.Also our approach does in general not yield explicit formulas.
We equip P d 1 with a Riemannian metric d( ; ).Let P denote the natural projection of a subset in R d nf0g to P d 1 .A matrix A 2 R d d de nes a map PA : P d 1 nPkerA !PImA as A maps one-dimensional subspaces into one-dimensional subspaces (or to 0).In homogeneous coordinates this means = PA i = x]; Ax 6 = 0; = Ax], where we have taken the usual equivalence relation on R d n f0g given by x y i 9 6 = 0 : x = y and x] denotes the equivalence class of x.With these remarks the associated system to (1) is given by (0) = 0 2 P d 1 ; u 2 U N ( 0 ): Here U N ( 0 ) denotes the set of admissible control sequences for 2 P d 1 , i.e. those control sequences u 2 U N such that (t; u)x 0 6 = 0 for all t 2 N, whenever Px 0 = 0 .The solution of (6) corresponding to an initial value and a control sequence u is denoted by '( ; ; u).In order to be able to use the results obtained in 24] we assume that the map A and the sets U Ũ R m satisfy: de nition of the forward orbit of a point given by O + ( ) := f 2 P d 1 j 9t 2 N; u 2 U t such that = '(t; ; u)g : The following concept introduces regions of approximate controllability in P d 1 .A control set is a set D P d 1 satisfying (i) D clO + ( ) for all 2 P d 1 .
(iii) D is a maximal set (with respect to inclusion) satisfying (i).
It is also possible to consider control sets with empty interior, but for the purposes of this paper this is unnecessary.Control sets have been studied in 1] and 24] and the references therein.We now present some of the relevant facts.
If we assume that ( 6) is forward accessible, i.e. intO + ( ) 6 = ; for every 2 P d 1 , then there exists a unique invariant control set in P d 1 , i.e. a unique set C satisfying cl C = cl O + ( ), 8 2 C. C is closed, connected and has nonempty interior.An important subset of a control set D is its core de ned by core(D) := f 2 D j int Ô ( ) \ D 6 = ; and int Ô+ ( ) \ D 6 = ;g : Here Ô ( ) denotes the points 2 P d 1 for which there exist t 2 N, u 0 2 intU t such that '(t; ; u 0 ) = and the map u 7 !'(t; ; u) has full rank in u 0 .Under these conditions ( ; u 0 ) is called a regular pair.A control u 2 int U t is called universally regular if ( ; u) is a regular pair for all 2 P d 1 .By 20, Corollaries 3.2 & 3.3] forward accessibility is equivalent to the fact that the set of universally regular control sequences is open and dense in U t for all t large enough.Ô+ ( ) is de ned by 2 Ô+ ( ) i 2 Ô ( ).
To the invariant control set C we may associate a set of Floquet exponents by Fl (C) := f 1 t log j j j 2 ( (t; u)); u 2 U t ; PGE( ; u) core(C)g : where we use the convention log 0 = 1.Here ( (t; u)) denotes the spectrum of (t; u) and GE( ; u) denotes the generalized eigenspace of an eigenvalue 2 ( (t; u)) or in the case of complex the kernel of the (real) matrix (( I (t; u))( I (t; u))) d .
The meaning of the Floquet exponents becomes clear if we introduce the exponential growth rate of a trajectory which is measured by the Lyapunov exponent Clearly, (x 0 ; u) < 0 i the corresponding trajectory converges to 0 exponentially fast, as measures the exponential growth of a trajectory.Due to the linearity (in x) of system (1) 3 Main result It is an easy consequence of 24, Theorem 11.1] that the in mum of Fl (C) characterizes exponential null controllability as it may be seen that sup 2P d 1 inf u2U N ( ; u) = inf Fl (C) ; (7) where possibly both sides are equal to 1.In fact, the in mum of the Floquet spectrum over C also characterizes asymptotic null controllability and feedback stabilizability as the following main theorem states.
Theorem 2 Let (A),(B),(C) hold and assume that ( 6) is forward accessible, then the following statements are equivalent.
(ii) System ( 1) is feedback stabilizable with a piecewise constant feedback F.
(iii) System ( 1) is exponentially feedback stabilizable with a piecewise constant feedback F.
PROOF.(i) ) (iv): Pick a point 0 2 core(C).By 24, Lemma 10.1] there exists a time T such that for every point 2 C there exists a control sequence u 2 U T with '(t ; ; u ) = 0 for some t T. By the boundedness of kA(u)k, u 2 U from above we can conclude that for all x 2 R d with Px = the estimate k (t ; u )xk Kkxk is valid for some constant K > 0 independent of x.Also, asymptotic null controllability implies that there exists a time t 0 > 0 and a control function u 0 2 U t 0 such that k (t 0 ; u 0 )x 0 k 1 2K kx 0 k ; for all x 0 2 R d with Px 0 = 0 .Now denote x 1 := (t 0 ; u 0 )x 0 .If x 1 = 0 it follows that inf Fl (C) = 1 and we are done.Otherwise by invariance of C 1 := Px 1 2 C. Choose u 1 2 U t 1 steering to 0 in time t 1 T. Concatenating u 0 and u 1 we obtain a control u 2 U t 0 +t 1 satisfying '(t 0 + t 1 ; 0 ; u) = 0 and k (t 0 + t 1 ; u)x 0 k 1 2 kx 0 k : This implies 1 (t 0 +t 1 ) log 2 2 cl Fl (C) and the assertion follows.(iv) ) (iii) follows from Theorem 11 below, while (iii) ) (ii) and (ii) ) (i) are immediately clear.a numerical point of view.We circumvent these problems by introducing approximations of (1) by invertible systems.We consider sequences fU n g n2N of compact sets satisfying for each n 2 N the following condition: where int U denotes the interior in the relative topology of U. One possible choice of such a sequence fU n g can be obtained via the following procedure.Let B(") := fu 2 U j det(A(u)) "g.Then we may choose " > 0 such that intB(") 6 = ;.Now de ne U n := B("=n) for n 1.
Assuming (8) we consider the approximating systems The following proposition states in what sense the systems (6 n ) are appropriate approximations of the original system.
Proposition 3 Let fU n g n2N be a family of sets satisfying ( 8) then (i) For every n 2 N system (6 n ) is forward accessible.
(ii) For every n 2 N system (6 n ) has a unique invariant control set C n . (iii (iv) System ( 6) is asymptotically null-controllable i there exists an n 0 2 N such that system (6 n ) is asymptotically null-controllable for all n n 0 .
PROOF.(i) By 20, Corollaries 3.2 & 3.3] forward accessibility of ( 6) and intU n 6 = ; imply that for t large enough (independently of n) there is a universally regular control sequence u n 2 intU t n .This implies that '(t; ; u n ) 2 int O + ( ) for all 2 P d 1 , which shows forward accessibility.
(ii) Using the control u n from part (i), let V (u n ) denote the sum of the eigenspaces of (t; u n ) corresponding to the eigenvalues of greatest modulus, i.e.
V (u n ) := M 2 ( (t;un));j j=r( (t;un)) GE( ; u n ) : Correspondingly, de ne W(u n ) := M 2 ( (t;un));j j<r( (t;un)) GE( ; u n ) : By 24, Proposition 6.7] there is a control set C n such that PV (u n ) core(C n ).We claim that C n is the unique invariant control set of (6 n ).Note that for 2 P d 1 n W(u n ) it holds lim k!1 d(P (t; u n ) k ; V (u n )) = 0 : As system (6 n ) is forward accessible and int PW(u n ) = ; this shows that PV (u n ) \ cl O + ( ) 6 = ; for all 2 P d 1 : ( One the one hand this shows that C n is invariant, as any trajectory with initial condition 0 2 C n can be steered back to core(C n ) such that the whole trajectory is contained in C n by maximality of control sets.On the other hand there is no other invariant control set C 0 as (9) shows that from some control set C 0 6 = C n it is possible to steer to C n contradicting the invariance of C 0 .(iii) As U n U n+1 U it follows that C n C n+1 C and inf Fl (C n ) inf Fl (C n+1 ) inf Fl (C).Thus it follows that lim n!1 inf Fl (C n ) exists and To prove the remaining inequality x c > inf Fl (C).Choose t 2 N, and a universally regular u 2 U t inv such that there exists 2 ( (t; u)) with PGE( ; u) core(C) and 1 t log j j < c.By universal regularity and the proof of (ii) we have PV (u) core(C).For 2 PV (u) and 2 PGE( ; u) we may by 24, Lemma 10.1 (ii)] choose a universally regular control v 2 U s inv such that = '(s; ; v) : As u 2 U t inv ; v 2 U s inv there exists an n 0 2 N such that u 2 intU t n ; v 2 intU s n holds for all n n 0 .By part (ii) we have V (u) core(C n ) and as the control v is available for (6 n ) the invariance of C n implies that 2 C n .On the other hand again using 24, Proposition 6.7] there is a control set D such that PGE( ; u) D. Now D \ C n 6 = ; and hence D = C n by the maximality of control sets.By de nition of the Floquet spectrum if follows that 1 t log j j 2 Fl (C n ) and so inf Fl (C n ) < c.As c > inf Fl (C) was arbitrary this completes the proof.
(iv) Clearly if (6 n 0 ) is asymptotically null-controllable then the same holds for all n n 0 and (6).It has already been shown in the implication (i) ) (iv) of Theorem 2 that asymptotic null-controllability of (6) implies that inf Fl (C) < 0. Hence for all n large enough inf Fl (C n ) < 0. Now (7) shows the claim.Note that the proof did not depend on the connectedness of U.
Finally, we have to point out in this section that inf Fl (C) < 0 does not imply that there exists a periodic sequence u such that the spectral radius satis es r( (t; u)) < 1.So that constructing a stabilizing feedback is not equivalent to the possibility of choosing a stable periodic system in the family (1).
However, for every T 2 sl(2; R), the group of real 2 2 matrices with determinant 1, there exist t 2 N and u 2 U t such that T = (t; u).This can be shown by a simple calculation.This means that in this case the projection ( 6) is completely controllable on P 1 .Thus C = P 1 and inf Fl (C) log 2 as (A(2; 1=2)) = f2; 1=2g.
Note also that the question, whether there exists a stable periodic system in the family ( 1) is in general algorithmically undecidable, as has been shown in 2], whereas the problem of approximating inf Fl (C) has an algorithmic solution as will be shown in the remainder of this article.

Construction of the Feedback
In this section we will give a constructive approach for the calculation of the exponentially stabilizing feedback for system (1).It is based on a dynamic programming approach, using the fact that optimal exponential growth rates can be approximated by discounted values along trajectories.
The construction of the feedback is related to the following optimal control problem: De ne the function q : P d 1 U ! R f 1g by q( ; u) := Note that q and thus also the sum over q are bounded from above.
v 0 ( ) := inffJ 0 ( ; u) j u 2 U N g is negative for all points in the projective space.v 0 may be approximated by value functions v corresponding to the following -discounted yield: e t q('(s; ; u); u(s)) ; u 2 U N ( ) ; 1 else.
We will also consider the value functions corresponding to the approximations U n , given by v ;n ( ) := inf Note that the series in the de nition of J is divergent i the partial sums tend to 1, and that by assumption (8) we have inf 2P d 1 v ;n ( ) > 1. Theorem 5 Consider system (1) and assume that its associated system ( 6) is forward accessible, then it holds that Furthermore, it holds for all n 2 N, that PROOF.The equalities on the right hand side in each statement follow from (7).
We now obtain an upper bound for lim sup !0 max 2P d 1(1 e )v ;n ( ).The same argument can be applied to v and this case is therefore omitted.Fix n 2 N and choose 0 2 core(C n ).By 24, Lemma 10.1 (ii)] there exists a time T 0 2 N such that every point 2 P d 1 can be controlled to 0 in a time t = t( ) T 0 by some control u = u( ).Using this u it follows from Bellman's principle of optimality that v ;n ( ) t 1 X s=0 e s q('(s; ; u); u(s)) + e t v ;n ( 0 ) and hence there exists "( ) such that for all 2 P d 1 we have (1 e )v ;n ( ) (1 e )v ;n ( 0 ) + "( ) where "( ) ! 0 as !0. By 23, Corollary 3.5] it holds that (1 e )v ;n ( 0 ) v 0;n ( 0 ) + "( ) where again "( ) ! 0 as !0. Together this implies lim sup The proof of the rst statement now follows from 23, Theorem 4.9], where convergence of v to v 0 on core(C) is shown.Now assume lim inf !0 max 2P d 1 v ;n ( ) = < max 2P d 1 v 0;n ( ), then 23, Proposition 3.8] and the boundedness of q on P d 1 U n yield a contradiction.
using optimal feedbacks for the discounted problem.We will now show that this is indeed the case.We construct a feedback as follows.
De nition 6 De ne F ;n : P d 1 !U n by the following procedure: For each 2 P d 1 choose a value u 2 U n such that q( ; u) + e v ;n ('(1; ; u)) becomes minimal and let F ;n ( ) := u.
The function F ;n will in general not be unique; nevertheless the existence of a value F ;n ( ) with the desired properties is always guaranteed by the continuity of q, v ;n and u 7 !'(1; ; u) and the compactness of U n .Denote the solution of the system using F ;n by '( ; ; F ;n ).It is a straightforward calculation to show that this feedback law is indeed an optimal control strategy for v ;n , i.e. it holds that J ( 0 ; F ;n ) := 1 X s=0 e s q('(s; 0 ; F ;n ); F('(s; 0 ; F ;n ))) = v ;n ( 0 ) : It turns out, however, that this feedback is also exponentially stabilizing.
Theorem 7 Assume that ( 1) is asymptotically null-controllable and ( 6) is forward accessible, then there exists an n 0 2 N and a 0 > 0 such that for all n n 0 and 0 < 0 the feedback F : R d !U n given by F(0) = u 0 for some arbitrary u 0 2 U n and F(x) = F ;n ( ) i Px = (11) exponentially stabilizes system (1).
PROOF.Let n 0 be such that inf Fl (C n 0 ) < 0 and choose 0 such that max 2P d 1 v ;n 0 ( ) < 0 for all 0 < < 0 .Then inf Fl (C n ) < 0 and max 2P d 1 v ;n ( ) < 0 for all (n; ) with n n 0 and 0 < 0 .Choose n n 0 ; 0 < 0 .Denote the exponential growth rate of an initial condition x 0 under the feedback F by (x 0 ; F).For any initial condition x 0 6 = 0 we have (1 e )J ('(t; Px 0 ; F ;n ); F ;n ) max 2P d 1 (1 e )v ;n ( ) < 0 ; where we used 23, Proposition 3.8] and the fact that q is bounded on P d 1 U n .To complete the proof it su ces to show the existence of a constant M > 1 such that kx(t; x 0 ; F)k Me t , 2P from (10) the existence of a T such that 1 t( ) t( ) X s=0 q('(s + t; ; F ); F ('(s + t; ; F ))) + " ; for all 2 P d 1 and some t( ) T. By induction and boundedness of A(U) exponential stability follows.

A Numerical Construction of the Feedback
Usually, it will not be possible to calculate v ;n explicitly.Instead we assume that we are given a numerical approximation to this optimal value function.From now on assume we have xed a compact control range U n U inv which approximates our original control problem to a desired accuracy.The main implication of this is the existence of a constant M q such that jq( ; u)j M q for all 2 P and all u 2 U n .
A numerical approximation of v ;n can be obtained as in 11]: Parameterizing P d 1 in a suitable way we obtain a transformation of the problem to some subset R d 1 on which we have to solve a discrete Hamilton-Jacobi-Bellman equation.The solution of this equation can be approximated on a grid covering where we look for a solution which is piecewise linear on each element of the grid.This solution can be calculated iteratively and, using the inverse of the parameterization, gives an approximation of v ;n on P d 1 .We denote this piecewise linear numerical approximation by ṽ .Using the results from 12] ṽ may be calculated in such a way that ṽ ( ) = inf u2Un fq( ; u) + e ṽ ('(1; ; u))g + ( ) (12) where j ( )j < for all 2 P d 1 and kṽ v ;n k 1 < ": From 12] it follows that > 0 and " > 0 can be made arbitrarily small using a suitable grid; furthermore ṽ is Hoelder continuous, i.e. it satis es jṽ ( 1 ) ṽ ( 2 )j Kd( 1 ; 2 ) where 2 (0:1] is an appropriate constant.Throughout this section K > 0 will denote several appropriate constants.Note that also v ;n is Hoelder continuous, see 10].
We will now use ṽ in order to construct an approximately optimal feedback.
De nition 8 De ne F : P d 1 !U n as follows.For any point 2 P d 1 choose a value q( ; u) + e ṽ ('(1; ; u)) becomes minimal and let F( ) := u.
The feedback as constructed in De nition 8 does not possess any regularity properties, in particular it will in general be discontinuous.However, it is possible to approximate this feedback by a piecewise constant function which still yields approximately optimal trajectories.
Proposition 10 Let fV j j j = 1; : : : ; Jg be a family of disjoint sets with S J j=1 V j = P d 1 and supfd( ; ) j ; 2 V j g for all j = 1; : : : ; J. De ne a feedback law F by F j V j F( j ) for arbitrary (but xed) points j 2 V j and all j = 1; : : : ; J and F from De nition 8.
Note that the previous proposition can in particular be used for the construction of piecewise constant feedbacks by imposing further regularity conditions on the V j .On way to obtain such sets is by using a partition (e.g.some triangulation) of the d dimensional unit sphere and then by identifying P d 1 with one hemisphere.Observe that the construction of the piecewise constant map can also be based on the feedback F ;n from De nition 6.The following theorem now states the main existence result for piecewise constant feedbacks.
Theorem 11 Consider system (1) and assume its associated projection ( 6) is forward accessible.Assume furthermore that inf Fl (C) < 0. Then there exists a piecewise constant feedback law F : P d 1 !U n such that J ( 0 ; F ) < 0 for all 0 2 P d 1 , and the map F : R d !U n given by F(0) = u 0 for u 0 2 U n arbitrary and F(x) = F ( ) i Px = (14) de nes an exponentially stabilizing piecewise constant feedback.
PROOF.From inf Fl (C) < 0, Theorem 7 and Proposition 10 the existence of F with the proposed properties follows.Hence we obtain 1 X s=0 e s q('(s + t; ; F ); F ('(s + t; ; F ))) < c for some value c < 0 and all t 2 N. Thus the assumptions of 23, Proposition 3.8] are satis ed and we obtain (x 0 ; F ) < c for all x 0 2 R d n f0g.As in the proof of Theorem 7 this implies exponential stability of the closed loop system on R d .
For semi-linear systems whose projection satis es a controllability assumption we have shown that open loop asymptotic null controllability and exponential feedback stabilizability via piecewise continuous maps is equivalent.For these feedbacks no explicit formula has been obtained and we also do not expect that a simple representation exists.Rather a numerical procedure for their construction has been presented.
Finally let us brie y comment on the robustness of the proposed stabilization scheme.One reason why feedback laws are preferred to open loop controls is that one expects some robustness of the stabilization against small errors or perturbations.
Unfortunately, F ;n as well as F and F are in general discontinuous, hence continuous dependence on the initial value will not hold for the closed loop system and thus not yield the desired robustness result.Nevertheless it is possible to show that the feedback controlled trajectories are robust in the sense that they remain approximately optimal if the system is subject to small perturbations using discrete-time versions of Proposition 5.2 in 14].
Proposition 9 Let F : P d 1 !U n be a feedback law obtained from De nition 8. Then the