Asymptotic methods in the theory of differential equations with discontinuous and multi-valued right-hand sides

Abstract

Zaremba [1] and Marchaud [2] carried out the first investigations of differential equations with a multivalued right-hand side. The authors of these works generalized the results of the theory of differential equations to a more general case, namely, to differential equations with a multi-valued right-hand side. Marchaud and Zaremba extended the class of admissible solutions to continuous functions and, instead of the ordinary derivative, used generalized concepts of derivative, i.e., contingencies (paratingencies), which are multi-valued mappings.For the next 25 years there were no work~ devoted to this problem because of the absence of applications. At the beginning of the 1960s, Wazewski [3,4] and Filippov [5, 6] published a series of works, in which they obtained fundamentally new results concerning the existence and properties of solutions of differential equations with a multi-valued right-hand side (differential inclusions). One of the most important results was the establishment of a relationship between differential inclusions and optimal control problems, and this led to rapid development of the theory of differential inclusions.Indeed, it is customary to write the equation describing the motion of a controlled object as the differential equationwhere u E U E comp (R '~) is a control vector. Under quite general assumptions, the controlled system (1) is equivalent to the differential inclusion e F(t, x(t)),where F(t, x) = f(t, x, U).h solution of the differential inclusion (2) means an absolutely continuous function x(t) that satisfies (2) for almost all t. The investigation of differential inclusions requires the study of the properties of multi-valued functions, i.e., the development of the whole mathematical apparatus for multi-valued functions.The reader can find an extensive bibliography concerning these investigations, e.g., in [7][8][9]. Suppose that comp (R '~) is the collection of all nonempty compact subsets of the space R '~ and cony (R '~) is the collection of all nonempty convex compact subsets of R '~. We denote by S~(a) and St(A) the closed r-neighborhood of the point a and the set A, respectively, by p(A,B) = min lira-bll aEA,bEB the Euclidean distance between the sets A, B E comp (1~'~), by ,8(A, B) = max p(a, B) aEA the semi-distance of the set A from the set B, and by h(A, B) = max{~(A, B),~(B, A)} the Hausdorff distance between the sets A and B.

On Existence of Limit Occupational Measures Set of a Controlled Stochastic Differential Equation

“…Note that singularly perturbed problems of control and optimization have been considered in both deterministic and stochastic literature (see [1], [2], [3], [4], [5], [6], [7], [8], [9], [10], [12], [17], [18], [19], [20], [21], [22], [24], [25], [26], [27], [28], [29], [30], [31], [32], [33], [35], [36], [37], [38], [39], [41], [42], [43], [44], [45], [46], [48], [50], [51], [53] and references therein). Singularly perturbed CSDE, in particular, have been studied in [2], [3], [12], [32], [33], and [38], where earlier references can also be found.…”

Stability and Optimality of Solutions to Differential Inclusions via Averaging Method

“…In Plotnikov [95], a version of the first Bogoliubov theorem was obtained for differential inclusions with periodic upper semi-continuous right-hand side.…”

“…In the same paper [95] Plotnikov also considered some special cases of upper semi-continuous differential inclusions, for which it is possible to establish estimates of the form (7). For example, he considered piecewise-continuous Lipschitzian set-valued maps of the form…”

On Average Control Generating Families for Singularly Perturbed Optimal Control Problems with Long Run Average Optimality Criteria

“…Introduction and preliminaries. Problems of optimal control of singularly perturbed (SP) systems have been studied intensively in both deterministic and stochastic settings (see [2], [8], [17], [18], [23], [26], [28], [29], [33], [41], [52], [55], [56], [58], [61], [63], [68], [67], [65], [74], [75], [77] for a sample of the literature). Originally, the most common approaches to SP control systems, especially in the deterministic case, were related to an approximation of the slow dynamics by the solutions of the systems obtained via equating of the singular perturbations parameter to zero, with further application of the boundary layer method (see [64], [73]) for an asymptotical description of the fast dynamics.…”

“…Various averaging type approaches allowing a consideration of more general classes of SP problems, in which the optimal and near optimal controls take the form of rapidly oscillating functions and in which equating of the small parameter to zero does not lead to a right approximation, were studied in [1], [2], [6], [7], [8], [9], [10], [18], [19], [20], [27], [28], [33], [34], [38], [39], [41], [42], [44], [51], [52], [68], [67], [70], [75] (see also references therein). This research lead to a good understanding of what the "true limit" problems, optimal solutions of which approximate optimal solutions of the SP problems, are.…”