Ordinary differential inclusions with internal and external perturbations (Q5951113)

Let \(\mathbb{R}^ n\) be the space of column \(n\)-vectors with Euclidean norm \(|\cdot|\); \(\text{comp}[\mathbb{R}^ n]\) be the set of all non-empty bounded closed subsets of \(\mathbb{R}^ n\); \(B[u,r]\) be the closed ball with center \(u\) and radius \(r>0\). For \(V\subset \mathbb{R}^ n,\) \(\operatorname {co}V\) denote the convex hull of the set \(V\). Let \(K([a,b]\times[0,\infty))\) be the set of all functions \(\eta:[a,b]\times[0, \infty)\to[0,\infty)\) that satisfy the following properties: \(\eta(t,\delta)\) is a measurable function for each \(\delta\in[0,\infty)\); for each \(\delta\in[0,\infty)\) there exists a function \(m_\delta:[a,b]\to[0,\infty)\) such that \(\eta(t,\tau)\leq m_\delta(t)\) for almost all \(t\in[a,b]\) and all \(\tau\in[0,\delta]\); \( \lim_{\delta\to 0+}\eta(t,\delta)=0\) and \(\eta(t,0)=0\) for almost all \(t\in [a,b]\). The authors consider the differential inclusion \[ x'\in F(t,x(t)), \qquad t\in[a,b],\tag{1} \] where \(F:[a,b]\times \mathbb{R}^ n\to\text{comp}[\mathbb{R}^ n]\) satisfies Carathéodory's conditions. Along with (1), they consider the differential inclusion \[ x'\in\text{co}F(t,x(t)), t\in[a,b].\tag{2} \] Let \(\eta(\cdot,\cdot), \eta_0(\cdot,\cdot)\in K([a,b]\times[0,\infty)).\) The differential inclusion \[ x'(t)\in F(t,B[x(t),\eta_0(t,\delta)])^{\eta(t,\delta)}, \quad t\in[a,b],\tag{3} \] is called a differential inclusion with internal and external perturbations, where \(\eta_0(\cdot,\cdot)\) is the radius of the internal perturbation and \(\eta(\cdot,\cdot)\) is the radius of the external perturbation. Each solution to (3) for a given \(\delta>0\) is called a \(\delta\)-solution to (1). In this paper, the authors establish a relationship between the sets of \(\delta\)-solutions to inclusion (1) and the solution set to inclusion (2).