Subaggregation with linear observation maps (Q1323061)

The differential inclusion \(x'(t)\in F(x(t))\) together with the observation map \[ H(x)= \begin{cases} \{y\in\mathbb{R}^ q\mid Ax\leq y\} &\text{for }x\in K\\ \emptyset &\text{for }x\not\in K\end{cases} \] is considered. It is assumed that \(F\) is an upper semicontinuous set-valued map with non- empty convex and compact values and \(K\) is a closed set such that \(F(x)\cap T_ K(x)\neq \emptyset\) for each \(x\in K\). The map \(A\) is linear. Using the viability theory, a comparison system \(y'(t)= g(y(t))\) is constructed. The comparison system satisfies the following condition: For each pair of initial values \(x_ 0\in K\) and \(y_ 0\in H(x_ 0)\) there exists a solution \(x\) to the above differential inclusion and a solution \(y\) of the comparison system with \(y(0)= y_ 0\) such that for all \(t\geq 0\), \(y(t)\in H(x(t))\).