Linear control systems with fuzzy parameter

Abstract

We consider linear control systems with undetermined parameters that are elements of a fuzzy set. For this system, we introduce the notion of pencil of trajectories and establish some of its properties. We also introduce the notion of fuzzy reachable set and prove that it is convex and compact.For the first time, the notion of fuzzy set was introduced in [1]. A differential equation with fuzzy initial conditions was considered in [2], and differential equations with fuzzy right-hand sides were studied in [3]. For equations of this type, the notions of solutions were introduced and theorems on their existence were proved.In the present paper, we consider a controlled linear differential equation with fuzzy parameters on the righthand side. The investigation of properties of this equation reduces to the investigation of a controlled differential inclusion with fuzzy right-hand side. We establish some properties of a fuzzy pencil of trajectories and a reachable set.Let Comp (R n ) (Conv (R n )) be the space of nonempty compact (and convex) subsets of the Euclidean space R n with the Hausdorff metricwhere A, B ∈ Comp (R n ), S r (x) is a ball of radius r ≥ 0 centered at a point x ∈ R n , and S r (A) = A+S r (0).Consider the differential inclusioṅwhere x ∈ R n is a phase vector, u(t) ∈ U (t) is a control vector, U (·) : R 1 → Conv (R m ) is a set-valued mapping, A(t), B(t), and C(t) are n × n, n × m, and n × k matrices, respectively, v ∈ R k is a fuzzy external action (noise), and v(t) ∈ V is a fuzzy set with characteristic function μ(x), μ(·) : R k → [0, 1], that satisfy the following conditions:Assumption 1. The matrices A(t), B(t), and C(t) are measurable on R 1 .2. There exist constants a > 0, b > 0, and c > 0 such that A(t) ≤ a, B(t) ≤ b, and C(t) ≤ c for almost all t ∈ R 1 .3. The set-valued mapping U (t) is measurable on R 1 .4. There exists a constant g > 0 such that U (t) ≤ g for almost all t ∈ R 1 .

Averaging of Fuzzy Integrodifferential Inclusions

“…The applications of fuzzy set theory can be found in many branches of science as physical, mathematical, differential equations and engineering sciences. Recently there have been new advances in the theory of fuzzy differential equations [12][13][14][15][16], fuzzy integrodifferential equations [17][18][19][20], differential inclusions with fuzzy right-hand side [21][22][23][24] and fuzzy differential inclusions [25][26][27] as well as in the theory of control fuzzy d i f f e r e n t i a l e q u a t i o n s [ 2 8 -3 0 ] , c o n t r o l f u z z y integrodifferential equations [31][32][33][34][35], controlfuzzy differential inclusions [36][37][38][39], and control fuzzy integrodifferential inclusions [40]. In works [41][42][43][44][45][46][47][48] various schemes of an average for the fuzzy differential equations and inclusions have been considered.…”

The Time-Optimal Problems for Controlled Fuzzy R-Solutions

“…The applications of fuzzy set theory can be found in many branches of regional, physical, mathematical, differential equations, and engineering sciences. Recently there have been new advances in the theory of fuzzy differential equations [34][35][36][37][38][39][40][41][42][43][44][45][46][47] and inclusions [48][49][50][51][52][53] as well as in the theory of control fuzzy differential equations [54][55][56][57] and inclusions [57][58][59].…”

Linear Control Problems of the Fuzzy Maps

“…The applications of fuzzy set theory can be found in many branches of regional, physical, mathematical, differential equations, and engineering sciences. Recently there have been new advances in the theory of fuzzy differential equations [36][37][38][39][40][41][42][43][44][45][46][47] and inclusions [43,[48][49][50][51][52] as well as in the theory of control fuzzy differential equations [53][54][55] and inclusions [56,57].…”