An application of quasivariational inequalities to linear control systems (Q2564177)

Let \(X\) be a nonempty convex subset of \(\mathbb{R}^n\), \(\Gamma:X\to 2^X\), \(\Phi:X\to 2^{\mathbb{R}^n}\). The paper considers the generalized quasivariational inequality problem \(\text{GQVI}(X,\Gamma,\Phi)\). Find \((\widehat x,\widehat z)\in X\times\mathbb{R}^n\) such that \[ \widehat x\in\Gamma(\widehat x),\;\widehat z\in\Phi(\widehat x)\text{ and }\langle\widehat z,\widehat x- y\rangle\leq 0\text{ for all }y\in\Gamma(\widehat x). \] In this paper, the author establishes an existence theorem for \(\text{GQVI}(X,\Gamma,\Phi)\) which is a more general version of his previous result. He derives an alternative result concerning the qualitative properties of the fixed point set of certain multifunction and gives an existence theorem of fixed points which lie on the relative boundary of the corresponding value. Then he gives an application to the linear control system \[ X'= A(t)X+ B(t)u,\quad u(t)\in\Omega, \] where \(T>0\), and \(A:[0, T]\to\mathbb{R}^{n\times n}\), \(B:[0, T]\to\mathbb{R}^{n\times m}\) are matrix functions, \(\Omega\subseteq \mathbb{R}^n\) a nonempty compact set. He finds the periodic solution whose final values lie on the relative boundary of the corresponding attainable set.