Deterministic and Stochastic Difierential Inclusions with Multiple Surfaces of Discontinuity (original) (raw)

The viability theorem for stochastic differential inclusions2

Stochastic Analysis and Applications, 1998

The aim of this paper is to combine two ways for representing uncertainty through stochastic differential inclusions: a "stochastic uncertainty", driven by a Wiener process, and a "contingent uncertainty", driven by a set-valued map, as well as to consider stochastic control problems with continuous dynamic and state dependent controls. This paper is also devoted to viability of a closed subset under stochastic differential inclusions, characterized in terms of stochastic tangent sets to closed subsets.

Weak Solutions of Stochastic Differential Equations with Discontinuous Coefficients

2001

In the present paper, we consider the stochastic differential equation dx(t) = f (t, x(t))dt + g(t, x(t))dW (t) (1) with Borel measurable functions f : R + × R d → R d and g : R + × R d → R d×d , introduce various definitions of weak solutions, and prove the corresponding existence theorems. We establish a theorem on the dependence of solutions on the initial conditions and right-hand sides and a theorem on the compactness of the set of probability laws of weak solutions. The existence theorem is proved under the only assumption that the mappings f and g are Borel measurable and locally bounded and without any additional conditions, in contrast with other well-known existence theorems [1-5]. Weak solutions of Eq. (1) are defined as weak solutions of some stochastic differential inclusion corresponding to Eq. (1). We use the following notation: R d×r is the space of real d × r matrices equipped with the Euclidean norm • ; R d×1 = R d ; R + = [0, +∞[, conv R d×r is the metric space of nonempty convex compact subsets of R d×r with metric κ(A, B) = max{β(A, B), β(B, A)}, where β(A, B) = sup a∈A inf b∈B a − b ; [B] α = x ∈ R d×r | inf y∈B x − y ≤ α is the α-neighborhood of a set B ⊂ R d×r ; µ is the Lebesgue measure on R + ; C R + , R d×r is the space of continuous functions a : R + → R d×r equipped with the metric (a 1 , a 2) = ∞ k=1 2 −k max 0≤t≤k a 1 (t) − a 2 (t) ∧ 1 , b 1 ∧ b 2 = min {b 1 , b 2 } ,

Existence Results for Stochastic Semilinear Differential Inclusions with Nonlocal Conditions

2011

We discuss existence results of mild solutions for stochastic differential inclusions subject to nonlocal conditions. We provide sufficient conditions in order to obtain a priori bounds on possible solutions of a one-parameter family of problems related to the original one. We, then, rely on fixed point theorems for multivalued operators to prove our main results.

Semicontinuity of multifunctions and a stochastic fluid oscillator

ZAMM, 2003

In this paper on stochastic differential inclusions with set-valued drift, we will consider the possible existence of strong solutions in the case of a kind of lower semicontinuous multifunctions as well as of upper semicontinuous ones. The dispersion matrix conditions are weakened in the classical way. To show existence results for such inclusions, we will use a locally Lipschitzean approximation of lower semicontinuous multifunctions. A set-valued stochastic system with an upper semicontinuous drift will be used to describe the motion of a fluid oscillator with small random perturbations.

Strong Solutions of Stochastic Differential Inclusions with Unbounded Right-Hand Side in a Hilbert Space

Differential Equations, 2018

In a separable Hilbert space, a stochastic differential inclusion with coefficients whose values are closed not necessarily convex sets is considered. Two existence theorems for strong solutions are proved. In the first theorem, the proof is based on the use of Euler polygonal lines; in the second, on the successive approximation method. Instead of the assumption that the coefficients of the inclusion are globally Lipschitz, which is traditional in such cases, some conditions that are less restrictive for the problems in question are used.

On the Existence of Solutions to Differential Inclusions with Nonconvex Right-Hand Sides

Siam Journal on Optimization - SIAMJO, 2007

We study the existence of solutions of differential inclusions with upper semicontinuous right-hand side. The investigation was prompted by the well known Filippov's examples. We define a new concept "colliding on a set". In the case when the admissible velocities do not "collide" on the set of discontinuities of the right-hand side, we expect that at least one trajectory emanates from every point. If the velocities do "collide" on the set of discontinuities of the right-hand side, the existence of solutions is not guaranteed, as is seen from one of the Filippov's examples. In this case we impose an additional condition in order to prove existence of a solution starting at a point of the discontinuity set. For the right-hand sides under consideration, we assume the following: whenever the velocities "collide" on a set S there exist tangent velocities (belonging to the Clarke tangent cone to S) on a dense subset of S. Then we prove existence of an ε-solution for every ε > 0. Under additional assumptions we can pass to the limit as ε → 0 and obtain a solution of the considered differential inclusion. Key words. differential inclusions with nonconvex right-hand side, existence of solutions; colliding on a set AMS subject classifications. 34A36, 34A60

Qualitative Properties of the Solution Set for Time-Delayed Discontinuous Dynamics

Set-Valued and Variational Analysis, 2019

This article presents a survey of several properties of the set of solutions for a differential inclusion involving a time-delayed component and with right-hand side parametrized by either an upper semicontinuous or lower semicontinuous multifunction. Our results include: existence of solutions, compactness and contractibility of the solution and reachable sets in the upper semicontinuous case, precompactness and connectedness of the solution and reachable sets in the lower semicontinuous case, regularity of the solution and reachable mappings with respect to parameters, and existence of solutions for a dynamic optimization problem.

Trajectories of differential inclusions with state constraints

Journal of Differential Equations, 2011

The paper deals with solutions of a differential inclusionẋ ∈ F (x) constrained to a compact convex set Ω. Here F is a compact, possibly non-convex valued, Lipschitz continuous multifunction, whose convex closure coF satisfies a strict inward pointing condition at every boundary point x ∈ ∂Ω. Given a reference trajectory x * (·) taking values in an ε-neighborhood of Ω, we prove the existence of a second trajectory x : [0, T ] → Ω which satisfies x − x * W 1,1 ≤ Cε(1 + | ln ε|). As shown by an earlier counterexample, this bound is sharp.