## 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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### References:

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