Impulsive Input-to-State Stabilization of an Ensemble

We consider an ensemble of trajectories generated by a linear differential equation subjected to disturbance and parameterized by the initial state. The scalar output of the system is the volume comprised by the states of the whole ensemble. Already the unperturbed dynamics is assumed to be unstable. In order to stabilize the system with unknown inputs in the ISS sense we design impulsive control actions based in the output signal and establish conditions under which the system possesses the ISS property under these impulsive actions.


Introduction
Mathematical modeling of multi-agents leads to different types of large-scale systems.One of them are so called ensembles, where the properties of constitutive dynamics for each agent changes continuously and can be parametrized by a scalar.This leads to a continuum of systems, see e.g., [1][2][3][4][5].Another situation is an ensemble of identical agents having different initial states taken from a nonempty subset of the state space of an agent, see [6].This generates a continuum of trajectories and leads again to an infinite dimensional dynamical system.The latter case will be considered in this paper.
The questions of controllability, observability, reachability were studied in many works, see, e.g., [2,7,8].The latter work develops ellipsoidal estimates of reachability sets for nonlinear dynamical systems with scalar impulsive actions and uncertain initial conditions.
Having an ensemble it is often desired in practical situations that its trajectories do not diverge too far from an equilibrium.This leads to stabilization problems, see [9].One possible approach is to apply impulsive actions as in [10,11].To this end we combine the classical control methods with Minkowski's geometric tools, see [12,13].
We consider the case where the unperturbed dynamics of the ensemble is not stable.Additionally, the dynamics is assumed to be subjected to a special class of disturbances.We assume that the volume V comprised by the states of all agents can be measured and consider it as the output.Based on this scalar value we are going to design impulsive actions which reduce the volume V instantaneously, if some critical value v * of V is achieved.The disturbance are assumed to be from class H α (see (6)) of signals with a non-vanishing volume α > 0. This is necessary in order to avoid the situation, when the volume V is arbitrary small, but some trajectories in it are unbounded.In this case a stabilization based on the measured scalar value V is not possible.
The main result of this work are the conditions on the impulsive action B (see below) and the critical volume value v * which guarantee the desired stability properties of the ensemble.

Notation and Preliminaries
Let R n be the Euclidean n-dimensional space with the standard scalar product (•, •), L(R n ) be the Banach algebra of linear operators with the norm A = sup x =1 Ax .A * denotes the adjoint operator to A. C(S n−1 ) denotes the Banach space of continuous functions f : conv R n denotes the space of nonempty compact and convex subsets of R n with Minkowski sum and multiplication by a non-negative scalar: for The Hausdorff metric on conv R n is defined by It is known [14] that the mapping i : For K = i(conv R n ), the set K is a linear semigroup C(S n−1 ).For a nonsingular linear operator B ∈ L(R n ), we define the operator B : For an operator A we define a family of mappings {S t } t∈R + by where f ∈ C(S n−1 ).One can prove that {S t } t∈R + is a C 0 -semigroup on C(S n−1 ) satisfying S t ≤ e γ t for t ≥ 0, where γ = 1 2 λ max (A * + A).Let A be the infinitesimal generator of S with domain D(A), then for any f ∈ K we have lim where (∇ p f , A * p) is the derivative of f in the direction A * p.Hence, D(A) ⊇ K.It is also clear that S t (K) ⊂ K for t ≥ 0. Consider a linear system with input there exists a unique solution x(t, t 0 , x 0 , u) to the Cauchy problem (1), defined for t ∈ [t 0 , T ] and satisfying the initial condition The input space is defined by The reachability set of the linear system (1) from a point x 0 ∈R n is defined by [7] X (t, t 0 , x 0 , U ) = {x(t, t 0 , x 0 , u) : u ∈ U t 0 ,t }.
Given the initial set X 0 ∈ conv R n (set of initial states), the solution to the ensemble (1) is denoted by It is known [7], that This equation can be written as an abstract differential equation on Let us introduce the notions of volume and mixed volume together with their properties following [15].V [X ] denotes the volume of X ∈ conv R n in the sense of usual Jordan measure.The non-trivial fact that for K 1 , . . ., K m ∈ conv R n and m ∈ N the volume Coefficients of this polynomial define the mixed volume M V so that we have where M V : and where Also we will use the following classes of comparison functions.
strictly increasing in the first argument with β(0, t) = 0 for any t ≥ 0 and decreasing in the second argument with lim t→∞ β(s, t) = 0 for any s ≥ 0.

Problem Statement
Consider the ensemble ∈ U 0,T and output y : R + → R + .Note that the overall system is not linear due to this output.Also note, even for a single initial value (when X 0 is a singleton) equations ( 5) generate an ensemble due to the ensemble of input signals u ∈ U .We assume that tr A > 0, which implies that the unperturbed ensemble (U = θ ) has an unbounded output for any initial set with non-zero volume.The disturbance U is assumed to be from the set for some fixed α > 0.
Our aim is to design an impulsive action on the base of the measured y(t) so that even under the disturbing input u the ensemble is stabilized.
In order to stabilize our system we denote 123 and look for a nonsingular bounded linear operator B : conv R n → conv R n with B < 1, which changes all elements of the ensemble instantaneously and contracts the volume y any time instant τ , when y reaches the critical value v * or, equivalently, if h X (τ,0,X 0 ,U ) ∈ D. In this case we have y(τ At all other time instants the ensemble is governed by (5).Our aim is to choose v * and B so that the resulting impulsive system becomes ISS in the following sense: Definition 2 The closed loop ensemble ( 5)-( 7) is said to be input-to-state stable (ISS) if for any given constants v * > 0 and α > 0 there exist β ∈ KL and γ ∈ K ∞ such that for any where .
Remark 1 This definition resembles the original definition of ISS from [16], however we consider only inputs from H α , which in particular excludes zero or arbitrary small disturbances in contrary to the usual ISS property.Note that for U = θ due to tr A > 0 the asymptotic stability for the resulting closed loop system is not possible, because Using h X (t) = h[t] and that i : X → h X | S n−1 is an isometric isomorphism, the closed loop system can be equivalently written as a hybrid one defined on C(S n−1 ) (see [17]): where h[t + ] = lim s→t+0 h[s], and study the ISS property of this system.

Main Result
We first establish the equation which governs the evolution of V .
Lemma 1 Let X (t) be the reachability set of the ensemble (5) where d + dt is the right side derivative, and V 1 is defined in (2). where 123 From (11) and by the monotonicity of V wrt set inclusions follows By the Steiner formula (3) we obtain For → 0 we have det( From ( 12) we obtain From (11) follows Dividing by > 0 and taking limit for → 0+ we obtain (10) by the continuity of the mixed volume.

Corollary 2 If we additionally assume inf
Proof Applying to (10) the inequality (4) and having By the comparison principle we obtain that 123 To state our main result we introduce the following notation Theorem 3 Let v * > 0 and B ∈ L(R n ) be such that δ ∈ (0, 1), β 0 ∈ (0, 1).Then the ensemble (5) satisfies the ISS property in the sense of Definition 2 with where k=1 ⊂ R + be the time sequence of the impulsive actions, that is h X (t k ) ∈ D. From tr A > 0, β 0 ∈ (0, 1) it follows that this sequence is strictly increasing.The dwell-time we denote by We rewrite (9) in the integral form taking into account that h U (s) ≤ U ∞ for all s ∈ [t k , t], we obtain with With help of (9), we obtain We estimate the dwell-times T k , k ∈ Z + by means of the Corollary 2 for T := t k+1 , t 0 := t k and having Hence for all k ≥ 1 it holds that 123 and from (17) we obtain the next estimate which is equivalent to Due to δ ∈ (0, 1) we can write Applying the Corollary 2 for T := t 1 and t 0 := 0 we get This implies From ( 9) and ( 16) we obtain With help of (20) it follows that for all k ≥ 1 we have Hence for all t ∈ (t k+1 , t k+2 ] from ( 16) we derive 123 Let t ∈ (t k+1 , t k+2 ], we have Because of > 1 this inequality is equivalent to Finally from (21) we obtain the estimate where the right hand side does not depend on k, which implies that this estimate is true for all t ∈ (t 2 , t * ), t * = lim k→∞ t k ≤ +∞.Let us prove that t * = ∞.If it is not true, then lim k→∞ T k = 0. From the definition of a norm it follows that By the monotonicity and positive homogeneity of the mixed volume functional we have and from (10) it follows that Taking into account, that )] = v * and passing to the limit in (23) for k → ∞ (T k → 0), we obtain v * ≤ β 0 v * , i.e. β 0 ≥ 1, which contradicts prerequisites of the theorem.

Discussion
We have considered linear differential equation generating an ensemble X (t) of solutions parameterized by initial states and subjected to unknown non-vanishing disturbances.It was assumed that the matrix of the system leads to a growth of the volume V [X (t)] of the ensemble in the state space.In this sense already the unperturbed system is not stable.To stabilize the system we have designed impulsive linear actions which are triggered when a prescribed critical value v * of V [X (t)] is reached.The overall impulsive system is nonlinear (because of V ).That is we deal only with such A that tr A > 0. Moreover we require that disturbances satisfy inf t∈R + V [U (t)] ≥ α > 0. The reason for the latter condition is that otherwise for V [X 0 ] = 0 and V [U (t)] = 0 it can happen that V [X (t)] = 0 for all t ≥ 0, but X (t) → ∞.That is the critical value v * will be never reached and impulsive stabilization will not work.
Let us note the dwell-time T k becomes smaller for larger α, see (18), which leads to the stabilization.Moreover the ISS-gain γ becomes smaller when α grows, see ( 15) and (18), which is a bit contra-intuitive at the first glance.We think that disturbances with large α preclude a stretching of the ensemble in separate directions and cause this interesting effect, which needs further investigations.
Note that in case of tr A ≤ 0 the relation between V [X (t)] and the norm of the state of an ensemble element needs further investigation.In particular if U = θ the value of V [X (t)] can be arbitrary small, while X (t) is arbitrary large.For example, for A ∈ R 2×2 with tr A < 0 and eigenvalues λ 1 λ 2 < 0 we have for t → ∞ that X (t) → ∞ whereas the Liuoville's theorem implies V [X (t)] → 0.

Example 1
Consider the system (5) with A = 2 −1 1 −1 and disturbances from H α with α = 0.5.Let us choose the critical value for the volume of ensemble to be v * = 1.For the stabilizing impulsive actions we choose B = 0.1I , where I stands for the identity map.For this choice of constituting parameters one can check that the conditions of Theorem 3 are satisfied with δ = 0.65 and the ISS property (8) is satisfied with γ (s) = 8.9s and β(s, t) = 4.5se −0.45t .