Relaxation Problems Involving Second-Order Differential Inclusions

Abstract

We present relaxation problems in control theory for the second-order differential inclusions, with four boundary conditions, $\ddot{u}(t)\in F(t,u(t),\dot{u}(t))$ a.e. on $[\mathrm{0,1}]$; $u(\mathrm{0})=\mathrm{0},\hspace{0.17em}\hspace{0.17em}u(\eta )=u(\theta )=u(\mathrm{1})$ and, with $m\ge \mathrm{3}$ boundary conditions, $\ddot{u}(t)\in F(t,u(t),\dot{u}(t))$ a.e. on $[\mathrm{0,1}];\hspace{0.17em}\hspace{0.17em}\dot{u}(\mathrm{0})=\mathrm{0},\hspace{0.17em}\hspace{0.17em}u(\mathrm{1})={\sum }_{i=\mathrm{1}}^{m-\mathrm{2}}\mathrm{‍}{a}_{i}u({\xi }_i)$, where , and $F$ is a multifunction from $[\mathrm{0,1}]\times {ℝ}^n\times {ℝ}^n$ to the nonempty compact convex subsets of ${ℝ}^n$. We have results that improve earlier theorems.