Set differential systems and vector Lyapunov functions
The paper is concerned with the stability of the trivial solution for the set differential system \(D_HU=F(t,U)\). Here, \(F\in C(\mathbb{R}_{+}\times K_c(\mathbb{R}^n)^N,K_c(\mathbb{R}^n)^N)\), \(K_c(\mathbb{R}^n)\) denotes the collection of all nonempty, compact and convex subsets of \(\mathbb{R}^n\) regarded as a metric space via the action of the Hausdorff metric, and \(D_H\) denotes the Hukuhara derivative (see the preliminaries in Section 2 of the paper). The paper focuses on two results (Theorems 3.2 and 4.1). The first one reads as: Assume that {\parindent=8mm \begin{itemize}\item[(i)]\(V\in C(\mathbb{R}_{+}\times K_c(\mathbb{R}^n)^N,\mathbb{R}_{+}^N)\), \[ (| V_1(t,U_1)-V_1(t,U_2)| ,\dots ,| V_N(t,U_1)-V_N(t,U_2)| )\leq AD^{\gamma}[U_1,U_2] \] (given by \((D(U_{11},U_{21}), \dots ,D(U_{1N},U_{2N}))\)), where \(A\) is a matrix with nonnegative elements; \item[(ii)]For \((t,U)\in \mathbb{R}_{+}\times K_c(\mathbb{R}^n)^N\), \(D^{+}V(t,U)\leq G(t,V(t,U))\), where \(G\in C(\mathbb{R}_{+}\times \mathbb{R}_{+}^N,\mathbb{R}^N)\); \item[(iii)]Suppose further that \(G(t,w)\) is quasi-monotone in \(w\) for each \(t\in \mathbb{R}_{+}\) and \(r=r(t,t_0,w_0)\) is the maximal solution of \(w^{\prime }=G(t,w)\), \(w(t_0)=w_0\geq 0\) existing for \(t\geq t_0\). \end{itemize}} Then \(V(t,U(t))\leq r(t)\), \(t\geq t_0\), where \(U\) is a solution of the set differential system above with \(U(t_0)=W_0\in K_c(\mathbb{R}^n)^N\). The second result is: Assume that {\parindent=8mm \begin{itemize}\item[(i)]\(G\in C(\mathbb{R}_{+}\times \mathbb{R}_{+}^N,\mathbb{R}^N)\), \(G(t,0)=0\) and \(G(t,w)\) is quasi-monotone in \(w\) for each \(t\in \mathbb{R}_{+}\); \item[(ii)]\(V\in C(\mathbb{R}_{+}\times S(\rho ),\mathbb{R}_{+}^N)\), where \(S(\rho )=\{U\in K_c(\mathbb{R}^n)^N:\sum_{i=1}^ND(U_i,0)<\rho \}\), \((| V_1(t,U_1)-V_1(t,U_2)| ,\dots ,| V_N(t,U_1)-V_N(t,U_2)| )\leq AD^{\gamma}[U_1,U_2]\), where \(A\) is a matrix with nonnegative elements and the function \(V_0(t,U)=\sum_{i=1}^NV_i(t,U)\) satisfies \(b(\sum_{i=1}^ND(U_i,0))\leq V_0(t,U)\leq a(\sum_{i=1}^ND(U_i,0))\), where \(a\), \(b\) are nonnegative, continuous and increasing functions with \(a(0)=b(0)=0\); \item[(iii)]\(D^{+}V(t,U)\leq G(t,V(t,U))\). \end{itemize}} Then, the stability properties of the trivial solution of \(w'=G(t,w)\) imply the corresponding stability properties of the trivial solution of the set-differential system. Two ingenious accompanying results (Lemmas 4.1 and 4.2) help to apply the general theorems to concrete problems. The paper is almost self-contained, the proofs are easy to read and quite interesting and an example closes the presentation.
scientific article; zbMATH DE number 2175471