Systems of control over set-valued trajectories with terminal quality criterion

Abstract

We consider the optimal control problem with terminal quality criterion in which the state of a system is described by a set-valued mapping, and an admissible control is a summable function. We describe an algorithm that approximates the admissible control function by a piecewise-constant function and prove theorems on the closeness of the corresponding trajectories and the values of quality criteria.The theory of set-valued mappings has been extensively developed since the late 1960s. In [1], Hukuhara introduced the notions of derivatives and integrals of set-valued mappings and studied the relationship between them. Later, differential equations with Hukuhara derivatives were considered in [2], various definitions of solutions were given and theorems on the existence of these solutions were proved in [3], and the applicability of certain averaging schemes to them was considered in [4,5].Equations with Hukuhara derivatives were used for the investigation of certain properties of an "integral funnel" of a differential inclusion in a Banach space in [6] and for the investigation of equations with fuzzy initial conditions in [7,8].In the present paper, we consider the problem of control over a process described by a linear differential equation with Hukuhara derivative and terminal quality criterion. This problem is significantly simplified if the control function is approximated by a piecewise-constant function. We present an algorithm for the construction of this approximate piecewise-constant control and prove the closeness of the corresponding trajectories and the values of quality criteria.Let Conv( ) R n be the space of nonempty compact and convex subsets of the Euclidean space R n with Hausdorff metric h(⋅ ⋅), . Consider the following control system described by a linear differential equation with Hukuhara derivative:n is a set-valued mapping that determines the state of the system, D X t h ( ) is the Hukuhara derivative [1], A t ( ) is an n n × matrix, F T (⋅) [ ] :, 0 → Conv( ) R n is the deviation of the system, and u(⋅) ∈ U ∈ Conv( ) R n is a control action.

Existence and Uniqueness Theorem for Set-Valued Volterra Integral Equations

“…de Blasi and F. Iervolino [1]. After that the properties of solutions of SDEs [2][3][4][5][6][7][8][9][10][11][12][13][14], the set-valued integrodifferential equations (SIDEs) [15], the impulse set-valued differential equations [9,11,16], set-valued control systems [12,17,18,19], the set-valued integral equations (SIEs) [20,21] and asymptotic methods [9,11,12,19] were considered. On the other hand, SDEs and SIEs are useful in other areas of mathematics.…”

Piecewise Constant Control Set Systems

“…de Blasi, F. Iervolino [8] started the investigation of set differential equations (SDEs) in semilinear metric spaces. This has now evolved into the theory of SDEs as an independent discipline: properties of solutions [1][2][3]5,36], the impulse equations [1,2,37], control systems [38][39][40][41] and asymptotic methods [1][2][3][42][43][44][45][46]. On the other hand, SDEs are useful in other areas of mathematics.…”

Full Averaging of Set Integrodifferential Equations

“…de Blasi, F. Iervolino [13] started the investigation of set differential equations (SDEs) in semilinear metric spaces. This has now evolved into the theory of SDEs as an independent discipline: properties of solutions , the impulse equations [6,7,41], control systems [42][43][44][45] and asymptotic methods [6][7][8][46][47][48][49][50]. On the other hand, SDEs are useful in other areas of mathematics.…”