Existence of optimal pairs for optimal control problems with states constrained to Riemannian manifolds (Q6576861)

The authors consider an \(n\)-dimensional and complete Riemannian manifold \(M\) with a Riemannian metric \(g\), the tangent space \(T_{x}M\) of \(M\) at \(x\), the tangent bundle \(TM=\cup _{x\in M}T_{x}M\), and the control system: \(\overset{. }{y}(t)=f(t,y(t),u(t))\), for a.e. \(t\in \lbrack 0,T]\), the control \(u(\cdot ) \) and state \(y(\cdot )\) satisfying the constraints: \(u(t) \in \Gamma (t,y(t))\), for a.e. \(t\in \lbrack 0,T]\), \(y(t)\in Q\), \(\forall t\in (0,T)\), \((y(0),y(T))\in \mathcal{S}\).\ Here \(T>0\), \(U\) is a metric space, \( f:[0,T]\times M\times U\rightarrow TM\) and \(\Gamma :[0,T]\times M\rightarrow 2^{U}\) (the class of all subsets of \(U\)), \(Q\subseteq M\) and \(\mathcal{S} \subseteq M\times M\). A pair \((u(\cdot ),y(\cdot ))\) is feasible if \(u(\cdot )\in \mathcal{U}=\{w:(0,T)\rightarrow \mathcal{U}\mid w(\cdot )\) is measurable\(\}\), \(y(\cdot )\) is absolutely continuous, and the equation and the constraints are satisfied. Given maps \(f_{0}:[0,T]\times M\times U\rightarrow \mathbb{R}\) and \(h:M\times M\rightarrow \mathbb{R}\), a pair \( (u(\cdot ),y(\cdot ))\) is said admissible if it is feasible and \(f_{0}(\cdot ,y(\cdot ),u(\cdot ))\in L^{1}(0,T)\), the authors introduce the cost functional with respect to the control system: \(J(u(\cdot ),y(\cdot ))=\int_{0}^{T}f0(t,y(t),u(t))dt+h(y(0),y(T))\), \(\forall u(\cdot ),y(\cdot )\in \mathcal{P}_{ad}\), the set of all admissible pairs. The optimal control problem is formulated as: Find \((\overline{u}(\cdot ), \overline{y}(\cdot ))\in \mathcal{P}_{ad}\) such that \(J(\overline{u}(\cdot ), \overline{y}(\cdot ))=inf_{(u(\cdot ),y(\cdot ))\in \mathcal{P} _{ad}}J(u(\cdot ),y(\cdot ))\). Assuming different hypotheses on the data, among which a Cesari-type property, and \(\mathcal{P}_{ad}\neq \varnothing \), the authors prove that the control system admits at least one optimal pair. For the proof, the authors consider a minimization sequence on which they prove estimates. The paper presents two examples of such problems.