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Relaxation problems involving second-order differential inclusions. (English) Zbl 1272.49025

Summary: 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 \([0, 1]\); \(u(0) = 0\), \(u(\eta) = u(\theta) = u(1)\) and, with \(m \geq 3\) boundary conditions, \(\ddot{u}(t) \in F(t, u(t), \dot{u}(t))\) a.e. on \([0, 1]\); \(\dot{u}(0) = 0\), \(u(1) = \sum^{m - 2}_{i = 1} a_iu(\xi_i)\), where \(0 < \eta < \theta < 1\), \(0 < \xi_1 < \xi_2 < \cdots < \xi_{m - 2} < 1\) and \(F\) is a multifunction from \([0, 1] \times \mathbb R^n \times \mathbb R^n\) to the nonempty compact convex subsets of \(\mathbb R^n\). We have results that improve earlier theorems.

MSC:

49J45 Methods involving semicontinuity and convergence; relaxation
49J21 Existence theories for optimal control problems involving relations other than differential equations
34A60 Ordinary differential inclusions
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