ON PRACTICAL STABILITY OF DIFFERENTIAL INCLUSIONS USING LYAPUNOV FUNCTIONS

. In this paper we consider the problem of practical stability for diﬀerential inclusions. We prove the necessary and suﬃcient conditions using Lyapunov functions. Then we solve the practical stability problem of linear diﬀerential inclusion with ellipsoidal righthand part and ellipsoidal initial data set. In the last section we apply the main result of this paper to the problem of practical stabilization.

1. Introduction. In many problems of practical importance it is necessary to investigate dynamic behavior of a system under state constrains on a finite time interval. Such problems are studied by methods of the practical stability theory. The main research technique in the theory of practical stability is the direct Lyapunov's method and its generalizations. Note that practical stability of solution does not imply stability in Lyapunov's sense, and conversely, unstable (in Lyapunov sense) solution can have good behavior on a finite interval. It means that Lyapunov functions being used in the practical stability problems do not obligatory satisfy the conditions of the second Lyapunov's method theorems.
Different problems concerning stability analysis for differential inclusions are studied in [1,3,5,10,13,11,18,20,21]. In these works Lyapunov function is used to investigate global, strong and weak stability. Converse Lyapunov theorems are proved in [2,5]. The stabilization problem on the basis of weak stability is considered in [20].
The concept of practical stability was introduced in [9,17]. In works [8,15,16] the second Lyapunov method and methods of stability theory have been developed with respect to the problems of practical stability. Sufficient conditions of practical stability have been obtained for different types of practical stability, in some cases proved the necessary conditions. In [6,7,8,12], the concept that studies the properties of maximum sets of initial conditions has been proposed and the effective numerical methods for different practical problems have been developed (for instance, for the charge beams optimization problem). This approach was effective in studying the practical stability problems of differential inclusions solutions. In [6,12] the topological properties of optimal set of initial conditions (compactness, boundary and interior properties) have been studied both for strong and weak practical stability. In the case of linear differential inclusions and convex phase 1978 VOLODYMYR PICHKUR constraints different techniques for describing such sets have been proposed (for example, support function, Minkowski function were used). In [19] similar results were obtained for discrete inclusions. But concerning practical stability of differential inclusions constructive generalization of the Lyapunov function method is still an open problem.
In this paper we study the problem of practical stability for differential inclusions. We prove the necessary and sufficient conditions using Lyapunov functions. Further we solve the practical stability problem of linear differential inclusion with ellipsoidal righthand part and ellipsoidal initial data set. In the last section we apply the main result of this paper to the problem of practical stabilization.
2. Necessary and sufficient conditions of practical stability. In this paper we introduce the following basic notations: R n is an n-dimensional Euclidean space; x, y is the usual inner product of x, y ∈ R n , x = x, x ; K r (a) is the ball in where Q is symmetric positive definite n × n-matrix, a ∈ R n , r > 0; M * is the transpose of n × m-matrix M ; intA, ∂A are respectively the set of inner points and the boundary, We consider differential inclusion where x ∈ R n is an n-dimensional vector of phase coordinate, (x, t) ∈ D, D ⊂ R n+1 is a bounded domain. A set-valued mapping F : D → conv(R n ) is measurable with respect to variable t and satisfies the Lipschitz condition Here L(t) is a positive integrable function, (x, t) ∈ D, (y, t) ∈ D, F (0, t) = 0, (0, t) ∈ D. The map F is integrably bounded. It means that there exists an integrable positive function λ(·) so that A multifunction Φ : [t 0 , T ] → comp(R n ) prescribes state constraints, graph of the mapping Φ belongs to D, 0 ∈ intΦ(t), t ∈ [t 0 , T ]. Let x(t, z, s) be a solution of (1) corresponding to the Cauchy condition x(s) = z, G 0 ⊂ R n . Definition 2.1. We say, that the zero solution of differential inclusion (1) is if and only if there exists a continuous function V : D → R 1 such that the following conditions take place: 1).
is a solution of the differential inclusion (1).
Proof. Necessity. We assume that the zero solution of differential inclusion (1) is The set G * is compact [6,12]. Therefore we may define a continuous function g : R n → R 1 having the following properties [6]: Here is the distance between x ∈ R n and ∂G * . We denote by Ω(y, t) the set of all solutions of the differential inclusion (1) [14]. Let The set X(t 0 , y, t) ∈ comp(R n ) and generates a continuous set-valued mapping with respect to variables y, t [14]. Consider a function It is clear that V (y, t) = min z∈X(t0,y,t) g(z). The function V (y, t) is continuous [4].
Let us show that the conditions 1 -3 of the theorem hold. In fact, if y ∈ G 0 , is an arbitrary solution of the differential inclusion (1). It is clear that Therefore min .
We admit there exists V (x, t) so that the conditions of the theorem take place. Taking any point . Thus definition 2.1 holds and the zero solution of (1) Following traditions the function V (x, t) in theorem 2.2 is called the Lyapunov function.
Remark 2. If we claim that F is upper semicontinuous set-valued function with respect to x instead of the Lipschitz condition (2), then the sufficiency of theorem 2.2 holds Corollary 1. Suppose that there exists a continuously differentiable function V : D → R 1 such that conditions (4), (5) hold and upper derivative due to differential inclusion (1) Then the zero solution of differential inclusion (1) Corollary 2. Let there exists a continuous function V : D → R 1 such that conditions (4), (5) take place and for any solution x(t) of the differential inclusion (1) the function V (x(t), t) does not increase on Then the trivial solution of differential inclusion (1) is {G 0 , Φ(t), t 0 , T } -stable.
Corollary 3. Suppose that there exists a continuously differentiable function V : D → R 1 such that conditions (4), (5) hold and Then the zero solution of differential inclusion (1) 3. Practical stability of linear differential inclusion. In this section we consider linear differential inclusion dx dt ∈ A(t)x + E(0, H(t)), where x ∈ R n is an n-dimensional vector of phase coordinate, A(t) is a continuous n × n-matrix, H(t) is a continuous symmetric positive definite n × n-matrix. We denote by Θ(t, s) a fundamental matrix of the linear system dx dt = A(t)x, normalized at the point s so that Θ(s, s) = I, where I is the identity n×n-matrix. A continuous multifunction Φ : [t 0 , T ] → conv(R n ) prescribes phase constraints, This inclusion implies 0 ∈ intG * [6,12]. Here G * is the maximum set of initial conditions under the phase constraints Φ(t), t ∈ [t 0 , T ]. It means that for arbitrary x 0 ∈ G * any suitable solution of differential inclusion (8) belongs to Φ(t) for all t ∈ [t 0 , T ]. Under the conditions of this section G * is the convex compact set [6,12]. Now we prove the following result.
an n×n-matrix Q(t) is a positive definite solution of the matrix differential equation q > 0, Q 0 is a symmetric positive definite n × n-matrix. Then the trivial solution of differential inclusion (8) Proof. We consider the function where R(t) = Q −1 (t). We have so that condition (4) holds.
Since 0 ∈ intΦ(t) the support function c(Φ(t), ψ) > 0, ψ ∈ S. From (9) it follows that for all ψ ∈ S, t ∈ [t 0 , T ]. But the function Q(t)ψ, ψ is the support function of the set Using support function properties we get Thus condition (4) is true.
As far as . Thus the matrix R(t) satisfies the matrix differential equation We observe that R(t) = R * (t) and max v∈E(0,H(t)) Using (12), (14), we get The Cauchy inequality implies Taking into account (15), we obtain Hence dV dt (8) ≤ 0 if V (x, t) = R(t)x, x ≥ 1. From corollary 3 of theorem 2.2 it follows that the trivial solution of differential inclusion (8) is For instance, if Φ(t) = K r(t) (0), r(t) > 0 is a continuous function on [t 0 , T ], then is completely equivalent to condition (9). Here λ max (Q(t)) is the maximum eigenvalue of Q(t).
Consider a particular case. Let the righthand part of differential inclusion (8) be time-independent. It means that A(t) = A, H(t) = H, t ∈ [t 0 , T ], where A is n × n -matrix, H is a symmetric positive definite n × n-matrix. In this case the following statement is true.
an n × n-matrix Q is positive definite and satisfies the matrix equation and Q − Q −1 0 is a nonnegative definite matrix. Here Q 0 is a symmetric positive definite n × n-matrix, G 0 = E(0, Q 0 ).
Then the trivial solution of differential inclusion (8) Proof. The proof of theorem 3.2 is similar to the proof of the previous theorem. Let us consider the Lyapunov function where R = Q −1 . From (17) it follows that R satisfies the matrix equation Equality (16) is equivalent to (4). From (18), (8) we get dV dt (8) Using support function properties we get G 0 = E(0, Q 0 ) ⊆ E(0, Q). But E(0, Q) = {x ∈ R n : V (x) ≤ 1} and condition (4) holds.

4.
On practical stabilization of differential inclusions. Given differential in- where, as it was above, (x, t) ∈ D, D ⊂ R n+1 is a bounded domain, a set-valued mapping F : D → conv(R n ) is measurable with respect to variable t, upper semicontinuous with respect to x, F (0, t) = 0, (0, t) ∈ D and satisfies (3). Further, G(t) is integrable n × m-matrix, u(x, t) is an m-dimensional control function, u(0, t) = 0. We assume that u(x, t) is integrably bounded on D, continuous with respect to variable x being measurable with respect to t. A multifunction Φ : [t 0 , T ] → comp(R n ) prescribes phase constraints, the graph of the mapping Φ belongs to D, 0 ∈ intΦ(t), t ∈ [t 0 , T ], G 0 ⊂ Φ(t 0 ). Suppose, that there exists a continuously differentiable function V : D → R 1 such that and The problem of {G 0 , Φ(t), t 0 , T } -stabilization for differential inclusion (19) consists of finding the admissible control function u(x, t) such that the zero solution to (19) C > 0. Then control function solves the problem of {G 0 , Φ(t), t 0 , T } -stabilization for differential inclusion (19).
Here Ω : [t 0 , T ] → comp(R n×n ) is a measurable integrably bounded multifunction. In other words, there exists an integrable positive function λ(·) so that Ω(t) ⊆ λ(t)B, t ∈ [t 0 , T ], where B is the unit ball in R n×n , R n×n is space of n × n-matrices with real components.