Inner approximations of the region of attraction for polynomial dynamical systems

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1 Inner approimations of the region of attraction for polynomial dynamical systems Milan Korda, Didier Henrion 2,3,4, Colin N. Jones October, 22 Abstract hal-74798, version - Oct 22 In a previous work we developed a conve infinite dimensional linear programming (LP) approach to approimating the region of attraction (ROA) of polynomial dynamical systems subject to compact basic semialgebraic state constraints. Finite dimensional relaations to the infinite-dimensional LP lead to a truncated moment problem in the primal and a polynomial sum-of-squares problem in the dual. This primal-dual linear matri inequality (LMI) problem can be solved numerically with standard semidefinite programming solvers, producing a hierarchy of outer (i.e. eterior) approimations of the ROA by polynomial sublevel sets, with a guarantee of almost uniform and set-wise convergence. In this companion paper, we show that our approach is fleible enough to be modified so as to generate a hierarchy of polynomial inner (i.e. interior) approimations of the ROA with similar convergence guarantees. Introduction Given an autonomous nonlinear system and a target set, the region of attraction (ROA) is the set of all states that end in the target set at a given time without leaving the state constraint set 5. The ROA is one of the principal sets associated to any dynamical system and goes by many other names in the literature (e.g., backward reachable set or capture basin [4]). In [6] we showed (in a controlled setting) that there is a genuinely primal conve characterization of the ROA. Optimization over system trajectories is formulated as optimization over occupation measures, leading to an infinite dimensional linear programming (LP) Laboratoire d Automatique, École Polytechnique Fédérale de Lausanne, Station 9, CH-5, Lausanne, Switzerland. {milan.korda,colin.jones}@epfl.ch 2 CNRS, LAAS, 7 avenue du colonel Roche, F-34 Toulouse; France. henrion@laas.fr 3 Université de Toulouse, LAAS, F-34 Toulouse; France 4 Faculty of Electrical Engineering, Czech Technical University in Prague, Technická 2, CZ-6626 Prague, Czech Republic 5 There are various modifications on this setup (e.g., one may consider asymptotic convergence instead of finite-time reachability, with or without constraints, in the presence of disturbances and/or uncertainty, or in a controlled setting); most of these modifications are amenable to the methods presented in this paper, sometimes with different qualitative results.

2 hal-74798, version - Oct 22 problem in the cone of nonnegative measures. Finite dimensional relaations of the dual of this problem then provide a converging sequence of outer approimations to the ROA. For a description of alternative techniques for numerical approimations of the ROA, please consult [5] or [6] and the many references therein. In this paper we show, within the same measure-theoretic framework, that there eists an infinite dimensional LP whose finite-dimensional relaations provide a converging sequence of inner approimations to the ROA. This paper can therefore be seen as a complement to [6]. To simplify our developments and to emphasize our contribution, we focus only on the uncontrolled setting. The main idea is to construct a converging sequence of outer approimations to the complement of the ROA. There are certain difficulties, topological in nature, associated with this approach. A careful distinction had to be made between trajectories leaving the constraint set and trajectories hitting its boundary. This then translates to a (sometimes subtle, but necessary) distinction between open and closed semialgebraic sets. Fortunately, the LP formulation proposed in [6] was fleible enough to allow for these modifications. Generally speaking, and consistently with our previous work [6], we believe that the main virtues of our approach are overall conveity, conceptual simplicity and compactness. Both primal and dual finite-dimensional relaations turn out to be linear matri inequalities (LMI), also called semidefinite programming (SDP) problems, with no tuning parameters besides the relaation order and no initialization data besides the defining ingredients of the problem. In addition, the inner approimations obtained are particularly simple they are given by a sublevel set of a single polynomial of a predefined degree. Therefore, an ROA approimation in analytic form can be readily obtained by solving a single LMI using freely available software (e.g., SeDuMi [2]). 2 Problem statement Consider the autonomous system ẋ(t) = f(t, (t)), (t) X R n, t [, T ] () with a given vector field f with polynomial entries f i R[t, ], i =,..., n, final time T >. The state trajectory ( ) is constrained to a nonempty open 6 basic semialgebraic set 7 X := { R n : g X () > }, (2) where the polynomial g X R[] is such that the set is compact 8. X := { R n : g X () } X 6 The requirement of the constraint set being open is merely technical, for this considerably simplifies the developments and the proofs. 7 For clarity of eposition we consider the constraint set given by a single superlevel set of a polynomial. The approach can, however, be straightforwardly etended to constraint sets defined by the intersection of finitely many polynomial superlevel sets. 8 Note that the closed semialgebraic set X = { : g X () } can be strictly larger than the closure of the open semialgebraic set X = { : g X () > }, consider in R e.g. g X () = ( 2 )(2 + ) 2. For a similar reason, note also that X bounded does not imply X bounded. Indeed, in R 2 with g X () = ( 2 2 2)(2 + ) 2 we have X = { : < } and X = { : } { : = 2}. 2

3 The vector field f is polynomial and therefore Lipschitz on the compact set X. As a result, for any X there eists a unique maimal solution ( ) to ODE (). The time interval on which this solution is defined contains the time interval on which (t) X. 2. Region of attraction (ROA) Given a final time T and an open bounded basic semialgebraic target set X T := { R n : g T () > } X, the region of attraction (ROA) is defined as { } X := X : ( ) s.t. ẋ(t) = f(t, (t)), () =, (T ) X T, (t) X, t [, T ]. (3) In words, the ROA is the set of all initial states from X for which the unique solution to () stays in X for all t [, T ] and ends in the target set at time T. hal-74798, version - Oct Complement ROA The idea to get inner approimations of the ROA X is to construct outer approimations of the complement ROA X c := X \ X. By continuity of solutions to (), the set X c is equal to where X c = { X : ( ) s.t. ẋ(t) = f(t, (t)) and is the complement of X T in X and t [, T ] s.t. (t) X and/or (T ) X c T }, X c T := { R n : g X (), g T () } X := { R n : g X () = }. In words, X c is the set of initial states that give rise to trajectories which do not end up in X T at time T and/or violate the state constraint at some point between and T. 3 Occupation measures In this section we introduce the concept of occupation measures and show how the nonlinear system dynamics can be equivalently described by a linear equation on measures. Notation We will use the following notation. The vector space of all signed Borel measures with support contained in a Borel set K is denoted by M(K). The support (i.e., the smallest closed set whose complement has a zero measure) of a measure µ is denoted by spt µ. The space of continuous functions on K is denoted by C(K) and likewise the space of continuously differentiable functions is C (K). The indicator function of a set K 3

4 (i.e., the function equal to one on K and zero otherwise) is denoted by I K ( ). The symbol λ denotes the n-dimensional Lebesgue measure (i.e., the standard n-dimensional volume). The integral of a function v w.r.t a measure µ over a set K is denoted by v() dµ(). K Sometimes we for simplicity omit the integration variable and/or the set over which we integrate if they are obvious from the contet. Now assume X and define the first hitting time of X as τ( ) := min { T, inf{t : (t ) X } }, (4) where ( ) denotes the unique trajectory starting from (which is well defined on the time interval [, τ( )]). Then we define the occupation measure associated to the trajectory starting from by µ(a B ) := τ( ) I A B (t, (t)) dt hal-74798, version - Oct 22 for all Borel 9 sets A B [, T ] X. The interpretation is that the occupation measure measures the time spent by the trajectory ( ) in subsets of the state space. The occupation measure enjoys the following important property: for any measurable function v(t, ) the equality τ( ) v(t, (t)) dt = [,T ] X v(t, ) dµ(t, ) (5) holds. In words, the time integral of a function evaluated along the trajectory ( ) is equal to the integral of the function w.r.t. the occupation measure associated to. Therefore, loosely speaking, all information about the trajectory ( ) is encoded by the occupation measure µ( ). Now suppose that the initial state is not a single point but that its spatial distribution is given by an initial measure µ M( X). Then we define the average occupation measure µ M([, T ] X) as µ(a B) := µ(a B ) dµ ( ). X Lastly, we define the final measure µ T M([, T ] X) by µ T (B) := I B ((T )) dµ ( ). X To derive an equation linking the three principal measures, consider a test function v C ([, T ] X) evaluated along a trajectory. Using the chain rule and equation (5) we 9 For brevity we drop the adjective Borel in the sequel. 4

5 obtain v ( τ( ), (τ( ) ) ) v(, ) = = = = τ( ) τ( ) [,T ] X [,T ] X d dt v(t, (t )) dt ( ) v t + grad v f(t, (t )) dt ( ) v + grad v f(t, ) dµ(t, ) t Lv(t, ) dµ(t, ) where the linear operator L : C ([, T ] X) C([, T ] X) is defined by hal-74798, version - Oct 22 v Lv := v t + grad v f. Integrating the above equation w.r.t. µ leads to the equation v(t, ) dµ T (t, ) v(, ) dµ () = Lv(t, ) dµ(t, ) [,T ] X X [,T ] X v C ([, T ] X), (6) which is a linear equation linking the measures µ, µ and µ T. Equation (6) is sometimes referred to as Liouville s equation. 4 Primal LP In this section we follow the approach developed in [6] and derive an infinite-dimensional linear programming (LP) characterization of the complement ROA X c. Certain sublevel sets of feasible solutions to the dual of this LP then yield inner approimations to the ROA X. The basic idea is to maimize the mass of the initial measure µ under the constraint that it is dominated by the Lebesgue measure, i.e., µ λ. System dynamics is captured by Liouville s equation (6) and state and terminal constraints are handled by suitable constraints on the support of the measures. The key idea is then to split the final measure in two measures such that each measure is supported on a suitable compact basic semialgebraic set. More eplicitly, we let µ T := µ T + µ 2 T with µ T M([, T ] X ) and µ 2 T M({T } Xc T ). That is, we require that spt µ T [, T ] X and spt µ 2 T {T } Xc T. The interpretation is that measure µ T models the trajectories that leave X, whereas measure µ 2 T models the trajectories that end in Xc T (i.e., not in X T ). These support constraints on the final measure(s) along with system dynamics enforce that the support of the initial measure µ must be contained in X. c Since there are no other constraints on µ besides µ λ, maimization of its mass should yield the restriction of the Lebesgue measure λ to X. c The constraint µ λ can be rewritten equivalently as µ + ˆµ = λ for some nonnegative slack measure ˆµ M(X). This is equivalent to requiring that w dµ + w dˆµ = w dλ 5

6 for all test functions w C( X). In addition, we can drop the time argument from the definition of µ 2 T since its time component is supported on a singleton. This leads to the following optimization problem: p = sup dµ s.t. v dµ T + v(t, ) dµ 2 T v(, ) dµ = Lv dµ v C ([, T ] X) w dµ + w dˆµ = w dλ w C( X) µ, µ, µ T, µ2 T, ˆµ spt µ [, T ] X, spt µ X, spt ˆµ X spt µ T [, T ] X, spt µ 2 T Xc T, (7) where the supremum is over the vector of nonnegative measures (µ, µ, µ T, µ 2 T, ˆµ ) M( X) M([, T ] X) M([, T ] X ) M(X c T ) M( X). hal-74798, version - Oct 22 Problem (7) is an infinite-dimensional LP in the cone of nonnegative measures. Indeed, the objective is linear, the first two constraints are linear equality constraints and the remaining constraints are conic constraints (the set of nonnegative measures supported on a given set is a positive cone in the vector space of all measures supported on the same set). The discussion leading to problem (7) is formalized in the following result. Theorem The optimal value of LP problem (7) is equal to the volume of the complement ROA X c, that is, p = λ(x c ). Moreover, the supremum is attained by the restriction of the Lebesgue measure to the complement ROA X c. Proof: Closely follows arguments in [6]. By definition of the relaed complement ROA, the unique trajectory ( ) associated to any initial condition X c either hits X at some t [, T ] or ends in XT c. Therefore for any initial measure µ λ with spt µ X X there eist an occupation measure µ, final measures µ T, µ2 T and a slack measure ˆµ such that the constraints of problem (7) are satisfied. One such measure µ is the restriction of the Lebesgue measure to X, c and therefore p λ(x). c Now we show that p λ(x c ). Take a vector of measures (µ, µ, µ T, µ2 T, ˆµ ) feasible in (7) and suppose that λ(spt µ \ X c ) >. Since any level set of a polynomial has a zero Lebesgue measure we have λ(x ) = and λ(spt µ \ (X c X )) = λ(spt µ \ X c ) >. By a superposition principle [, Theorem 3.2] using arguments of [6, Appendi A, Lemma 4], there eists a family of admissible trajectories of the ODE () starting from µ generating the occupation measure µ and the final measure µ T = µ T + µ2 T. However, this is a contradiction since spt µ \ (X c X ) X, which means that all trajectories starting from spt µ \ (X c X ) neither hit X nor end in X T c. Thus, λ(spt µ \ X) c = and so λ(spt µ ) λ(x). c Combining this with the constraint µ λ we get µ (X) = µ (spt µ ) λ(spt µ ) λ(x) c for any feasible µ. Therefore p λ(x) c and thus in fact p = λ(x). c 6

7 5 Dual LP In this section we derive a dual LP on continuous functions, prove the absence of a duality gap between the primal and dual LPs and relate feasible solutions to the dual to the indicator function of the complement ROA X c. By standard infinite-dimensional LP theory (see, e.g., [2]), the dual to LP (7) reads d = inf w() dλ() X s.t. Lv(t, ), (t,, u) [, T ] X w() v(, ) +, X v(t, ), v(t, ), XT c (t, ) [, T ] X w(), X, (8) hal-74798, version - Oct 22 where the infimum is over (v, w) C ([, T ] X) C( X). The intuition is that given X c the constraint Lv forces v to decrease along trajectories as long as it does not hit X or end in XT c. Because of the constraint v on [, T ] X {T } XT c we must have v(, ) on Xc. Consequently, w() on X. c This instrumental observation is formalized in the following Lemma. Lemma If Lv on [, T ] X, v on ([, T ] X ) ({T } XT c ) and w v(, )+ on X, then w on X. c Proof: Take X c and consider the first hitting time of X, τ := τ( ), defined by (4). By definition of X c the trajectory starting from will either hit X or end in XT c. Therefore (τ) ([, T ] X ) ({T } XT c ) and (t) X for t [, τ]. Therefore v(τ, (τ)), Lv(t, (t)), t [, τ] and so v(τ, (τ)) = v(, ) + τ Lv(t, (t)) dt v(, ) w( ). The following result is of key importance for subsequent developments. Theorem 2 There is no duality gap between primal LP problems (7) on measures and dual LP problem (8) on functions, that is, p = d. Proof: Here we only outline the basic steps; for a detailed argument in a similar setting see [6, Theorem 2]. Since the supports of all measures are compact, the initial measure is dominated by the Lebesgue measure and the final time is finite, we have µ ( X) λ( X) <, µ T ([, T ] X) = µ ( X) < and µ([, T ] X) T µ T ([, T ] X) <, where the last two inequalities follow by plugging in v(t, ) = and v(t, ) = t in Liouvillel s equation (6). Therefore p < and the feasible set of problem (7) is weakly-* bounded. Furthermore, the feasible set of (7) is nonempty since (µ, µ, µ T, µ2 T, ˆµ ) = (,,,, λ) is a trivial feasible point; therefore p <. The absence of a duality gap then follows from [2, Theorem 3.] using Alaoglu s theorem (see, e.g., [, Chapter 5]) and the weak-* continuity of the adjoint of the operator L. 7

8 Net, we establish our first convergence result. Theorem 3 There is a sequence of feasible solutions to problem (8) such that its w-component converges from above to I X in L norm and almost uniformly. Proof: Follows by the same arguments as Theorem 3 in [6]. 6 LMI relaations hal-74798, version - Oct 22 In this section we derive finite dimensional semidefinite programming (SDP) or linear matri inequality (LMI) relaations to the infinite dimensional LPs (7) and (8) and establish several convergence results relating these relaations to the infinite dimensional LPs and to the ROA. In what follows, R k [ ] denotes the vector space of real multivariate polynomials of total degree less than or equal to k. Derivation of the finite dimensional relaations is standard and the reader is referred to [6, Section 5] or to the comprehensive reference [9]; therefore we only highlight the main ideas. First of all, since the supports of all measures feasible in (7) are compact, these measures are determined by their moments, i.e., by integrals of all monomials (which is a sequence of real numbers when indeed in, e.g., the canonical monomial basis). Therefore, it suffices to restrict the test functions w() and v(t, ) in (7) to all monomials, reducing the linear equality constraints of (7) to linear equality constraints on the moments. Net, by the celebrated Putinar Positivstellensatz (see [9, ]), the constraint that the support of a measure is included in a given compact basic semialgebraic set is equivalent to the feasibility of an infinite sequence of LMIs involving the so-called moment and localizing matrices, which are linear in the coefficients of the moment sequence. By truncating the moment sequence and taking only the moments corresponding to monomials of total degree less than or equal to 2k we obtain a necessary condition for this truncated moment sequence to be the first part of a moment sequence of a measure with the desired support. This procedure leads to the primal SDP relaation of order k p k = ma (y ) s.t. A k (y, y, yt, y2 T, ŷ ) = b k M k (y), M k dx i (g X, y) M k (y ), M k dx (g X, y ) M k (yt ), M k d T (g X, yt ) M k d T ( g X, yt ) M k (yt 2 ), M k d T (g X, yt 2 ) M k d T ( g T, yt 2 ) M k (ŷ ), M k dx (g X, ŷ ) M k (t(t t), y), M k (t(t t), yt ) (9) where the notation stands for positive semidefinite and the minimum is over moment sequences (y, y, y T, y2 T, ŷ ) truncated to degree 2k corresponding to measures µ, µ, µ T, µ 2 T and ˆµ. The linear equality constraint captures the two linear equality constraints Please refer to [6] or, e.g., [3] for definitions of the various types of convergence relevant in this contet. 8

9 of (7) with v(t, ) R 2k [t, ] and w() R 2k [] being monomials of total degree less than or equal to 2k. The matrices M k ( ) are the moment and localizing matrices, following the notations of [9] or [6]. In problem (9), a linear objective is minimized subject to linear equality constraints and LMI constraints; therefore problem (9) is an SDP problem. The SDP problem dual to problem (9) turns out to be the sum-of-squares problem d k = inf w l s.t. Lv(t, ) = p(t, ) + q (t, )t(t t) + q 2 (t, )g X () w() v(, ) = p () + q ()g X () v(t, ) = p T () + q T (t, )t(t t) + r()g X () v(t, ) = p T 2() + q T 2()g X () q T 3()g T () w() = s () + s ()g X (), () hal-74798, version - Oct 22 where l is the vector of Lebesgue moments over X indeed in the same basis in which the polynomial w() with coefficients w is epressed. The minimum is over polynomials v(t, ) R 2k [t, ], w() R 2k [], over the polynomial r() and polynomial sum-of-squares p(t, ), q (t, ), q 2 (t, ), q (), p T (), p T 2(), q T (), q T 2(), q T 3(), s (), s () of appropriate degrees. The constraints that polynomials are sum-of-squares can be written eplicitly as LMI constraints (see, e.g., [9]), and the objective is linear in the coefficients of the polynomial w(); therefore problem () can be formulated as an SDP problem. Theorem 4 There is no duality gap between primal LMI problem (9) and dual LMI problem (), i.e. p k = d k. Proof: Follows by the same arguments based on standard SDP duality theory as Theorem 4 in [6]. Now we prove our main convergence results. Theorem 5 Let w k R 2k [] denote the w-component of a solution to the dual LMI problem () and let w k () = min i k w i (). Then w k converges from below to I X in L norm and w k converges from below to I X in L norm and almost uniformly. Proof: It follows from Theorem 3 and from the density of polynomials in the space of continuous functions on compact sets (for a detailed argument in a similar setting see [6, Theorem 5]) that w k and w k converge from above to I X c in L and almost uniformly on X, respectively. Therefore w k and w k converge from below to I X = I X c on X in the same manner. The net Corollary follows immediately from Theorem 5. Corollary The sequence of infima of LMI problems () converges monotonically from above to the supremum of the LP problem (8), i.e., d d k+ d k and lim k d k = d. Similarly, the sequence of maima of LMI problems (9) converges monotonically from above to the maimum of the LP problem (7), i.e., p p k+ p k and lim k p k = p. Proof: Follows the proof of Corollary in [6]. Monotone convergence of the dual optima d k follows immediately from Theorem 5 and from the fact that the higher the relaation 9

10 order k, the looser the constraint set of the minimization problem (). To prove convergence of the primal maima observe that from weak SDP duality we have d k p k and from Theorems 5 and 2 it follows that d k d = p. In addition, clearly p k p and p k+ p k since the higher the relaation order k, the tighter the constraint set of the maimization problem (9). Therefore p k p monotonically from above. Our last results establishes set-wise convergence of inner approimations to the ROA. Theorem 6 Let w k R 2k [] denote the w-component of a solution to the dual LMI problem () and let X k := { X : w k () < }. Then X k X, lim λ(x \ X k ) = and λ(x \ k=x k ) =. k hal-74798, version - Oct 22 Proof: Follows the proof of Theorem 6 in [6]. From Lemma we have w k () X for all X, and therefore I Xk I X. Since w k on X we also have w k I Xk on X and therefore w k I Xk I X on X. From Theorem 5, we have w k I X in L norm on X. Consequently, λ(x ) = I X dλ = lim w k dλ lim I Xk dλ = lim λ(x k ) X k X k X k lim λ( k k i=x i ) = λ( k=x k ). But since X k X for all k, we must have which proves the theorem. lim k λ(x k) = λ(x ) and λ( k=x k ) = λ(x ), 7 Numerical eamples In this section we present two numerical eamples. The primal problems on measures were modeled using Gloptipoly 3 [7] interfaced with the SDP solver SeDuMi [2]; this solver also returns the solution to the dual SDP relaation. In Section 7.3 we then investigate how tight low order approimations can be obtained. 7. Univariate cubic dynamics Consider the system given by ẋ = (.5)( +.5), the constraint set X = [, ], the final time T = and the target set X T = [.3,.3]. The ROA in this setup is X = [.5,.5]. Polynomial approimations to the complement ROA for degrees d {6, 2, 24, 28} are shown in Figure. As epected, the functional convergence of the polynomial to the discontinuous indicator function is rather slow. A slightly better convergence is observed in the volume error of the sublevel set approimation to the ROA documented in Table. The relatively slow convergence could be significantly improved if a tighter constraint set X was employed; see Section 7.3 below. Alternative polynomial bases (e.g. Chebyshev polynomials) would also allow tighter higher order approimations; see [8] for more details.

11 .2 d = 6.2 d = d = 24.2 d = 28 hal-74798, version - Oct Figure : Univariate cubic dynamics polynomial approimations (solid line) to the complement ROA indicator function I X c = I [,.5] + I [.5,] (dashed line) for degrees d {6, 2, 24, 28}. 7.2 Van der Pol oscillator As a second eample consider a scaled version of the uncontrolled reversed-time Van der Pol oscillator given by ẋ = 2. 2, ẋ 2 =.8 + ( 2.2) 2. We take T = and X T = { R n : 2.5} and X := { R n :.}. The ROA is bounded, having the characteristic Van der Pol shape. Plots of polynomial sublevel set approimations of degrees d {9, 2, 5, 8} are shown in Figure 3. We observe a relatively fast convergence to the ROA, which is also documented by the relative volume errors reported in Table 2. Figure 2 then shows a degree 8 polynomial approimation to the indicator function of the complement ROA. The coefficients were chosen so that the ROA fits within the bo [, ] 2.

12 Table : Univariate cubic dynamics relative error of the inner approimations to the ROA X = [.5,.5] as a function of the approimating polynomial degree. degree error.4 % 6.4 % 4.84 % 4.54 % hal-74798, version - Oct 22 Figure 2: Van der Pol oscillator degree 8 polynomial approimation to the indicator function of the complement ROA. Table 2: Van der Pol oscillator relative error of the inner approimation to the ROA X as a function of the approimating polynomial degree. 7.3 Low order approimations degree error 8.3 % 8.4 % 3.8 % 3. % In the eamples above, relatively high order polynomials had to be used to obtain tight approimations, which can limit subsequent applicability of the approimations. There are several ways to obtain low order approimations of similar quality. First of all, since the integral of a polynomial w is minimized over the constraint set X, it is desirable that X be a good outer approimation of the ROA. Of course, selecting X is possible only if it is an artificially specified outer approimation of the ROA, not a constraint set coming from physical requirements on the system. More importantly, notice that in problem () the system dynamics enters the constraints on the polynomial v(t, ), whereas the polynomial w() is only upper-bounding v(t, ) + for t =. Since the inner approimations are given by sublevel sets of w, it is possible and plausible to choose different degrees of w and v low for w and higher for v. Both techniques are illustrated in Figure 4; in Figure 4 (a) we consider the univariate cubic dynamics and we both shrink the constraint set X and choose low order w while keeping v of higher order. In Figure 4 (b) we consider the Van Der Pol oscillator, keeping the constraint set X unchanged and only selecting low order w. The inner approimations obtained are indeed significantly tighter for the given degrees (compare with Figures and 3). 2

13 d = 9.8 d = d = 5.8 d = hal-74798, version - Oct Figure 3: Van der Pol oscillator polynomial inner approimations (light gray) to the ROA (dark gray) for degrees d {9, 2, 5, 8} (a) Univariate cubic dynamics constraint set X = [.7,.7], deg w = 6 (deg v = 6).Volume approimation error 2.25 % (b) Van der Pol oscillator deg w = 8 (deg v = 8). Volume approimation error 5.46 %. Figure 4: Low order approimations to the ROA. Left: tighter constraint set X and low order w (compare with Figure ). Right: low order w only (compare with Figure 3). 8 Conclusion This paper presented an infinite dimensional conve characterization of the region of attraction (ROA) for uncontrolled polynomial systems, following the approach initiated in 3

14 hal-74798, version - Oct 22 our previous work [6]. Finite dimensional dual relaations yield a converging sequence of inner approimations to the ROA, thereby complementing the outer approimations of [6]. One of the virtues of the approach is its conceptual simplicity the resulting approimation is the outcome of a single SDP or LMI problem with no free parameters ecept for the relaation order. The approimations itself are also simple, given by sublevel sets of polynomials of predefined degrees. Nevertheless, this approach does not escape the curse of dimensionality indeed, whereas the number of variables of the LMI relaations grows polynomially with the relaation order, this number grows eponentially with the state dimension. Tailored structureeploiting SDP solvers could enable this approach to reach higher dimensions. In addition, a different choice of basis functions (e.g., Chebyshev polynomials rather than monomials) would improve numerical conditioning of the LMIs, allowing higher oder relaations to be computed. Future research directions include inner approimations in a controlled setting and the related problem of robust region of attraction / reachable set computation with either unknown but constant uncertainty or a time-varying disturbance. The cases of asymptotic region of attraction and maimum (controlled) positively invariant set computation are amenable to similar tools. References [] L. Ambrosio. Transport equation and Cauchy problem for non-smooth vector fields. In L. Ambrosio et al. (eds.), Calculus of variations and nonlinear partial differential equations. Lecture Notes in Mathematics, Vol. 927, Springer-Verlag, Berlin, 28. [2] E. J. Anderson, P. Nash. Linear programming in infinite-dimensional spaces: theory and applications. Wiley, New York, NY, 987. [3] R. B. Ash. Real analysis and probability. Academic Press, San Diego, CA, 972. [4] J. P. Aubin, A. M. Bayen, P. Saint-Pierre. Viability theory: new directions. Springer- Verlag, Berlin, 2. [5] G. Chesi. Domain of attraction; analysis and control via SOS programming. Lecture Notes in Control and Information Sciences, Vol. 45, Springer-Verlag, Berlin, 2. [6] D. Henrion, M. Korda. Conve computation of the region of attraction of polynomial control systems. LAAS-CNRS Research Report 2488, submitted to the IEEE Transactions on Automatic Control in August 22. URL: homepages.laas.fr/henrion/papers/roa.pdf [7] D. Henrion, J. B. Lasserre, and J. Löfberg. Gloptipoly 3: moments, optimization and semidefinite programming. Optimization Methods and Software, 24:76 779, 29. URL: homepages.laas.fr/henrion/software/gloptipoly. [8] D. Henrion, J. B. Lasserre, C. Savorgnan. Approimate volume and integration for basic semialgebraic sets. SIAM Review, 5: , 29. 4

15 [9] J. B. Lasserre. Moments, positive polynomials and their applications. Imperial College Press, London, UK, 29. [] D. G. Luenberger. Optimization by vector space methods. Wiley, New York, NY, 969. [] M. Putinar. Positive polynomials on compact semi-algebraic sets. Indiana University Mathematics Journal, 42: , 993. [2] I. Pólik, T. Terlaky, and Y. Zinchenko. SeDuMi: a package for conic optimization. IMA workshop on Optimization and Control, Univ. Minnesota, Minneapolis, Jan. 27. URL: sedumi.ie.lehigh.edu. hal-74798, version - Oct 22 5