Pontryagin's maximum principle for a fractional integro-differential Lagrange problem (Q6144084)

The author considers the optimal control problem: minimize \( \int_{a}^{b}g_{0}(t,y(t),u(t),v(t))dt\), subject to \((^{C}D_{a+}^{\alpha }y)(t)=g(t,y(t),u(t))+\int_{a}^{t}\frac{\Psi (t,s,y(s),v(s))}{ (t-s)^{1-\alpha }}ds\), \(u(t)\in M\subset \mathbb{R}^{k}\), \(v(t)\in N\subset \mathbb{R}^{l}\), a.e. \(t\in \lbrack a,b]\), \(y(a)=y_{0}\in \mathbb{R}\), where \(\alpha \in (0,1)\), \(g_{0}:[a,b]\times \mathbb{R}^{n}\times \mathbb{R} ^{k}\times \mathbb{R}^{l}\rightarrow \mathbb{R}\), \(g:[a,b]\times \mathbb{R} ^{n}\times \mathbb{R}^{k}\rightarrow \mathbb{R}^{n}\), \(\Psi :\Delta \times \mathbb{R}^{n}\times \mathbb{R}^{l}\rightarrow \mathbb{R}^{n}\), with \(\Delta =\{(t,s)\in \lbrack a,b]\times \lbrack a,b]:s<t\}\). Here \((D_{a+}^{\alpha }z)(\cdot )\) is the left-sided Riemann-Liouville derivative of order \(\alpha \) of the function \(z\in L_{n}^{1}\), the space of all summable functions \( z(\cdot ):[a,b]\times \mathbb{R}^{n}\), defined through: \((D_{a+}^{\alpha }z)(t)=\frac{d}{dt}(I_{a+}^{1-\alpha }z)(t)\), a.e. \(t\in \lbrack a,b]\), with \((I_{a+}^{\alpha }z)(t)=\int_{a}^{t}\frac{z(\tau )}{(t-\tau )^{1-\alpha }} d\tau \), the function \((I_{a+}^{1-\alpha }z)\) being absolutely continuous on \([a,b]\). \((^{C}D_{a+}^{\alpha }z)\) is the left-sided Caputo derivative of order \(\alpha \) of the function \(z\in C_{n}\), the space of all continuous functions from \([a,b]\) to \(\mathbb{R}^{n}\), such that the function \(z(\cdot )-z(a)\) has a Riemann-Liouville derivative, defined through \( (^{C}D_{a+}^{\alpha }z)(t)=D_{a+}^{\alpha }(z(t)-z(a))\), a.e. \(t\in \lbrack a,b]\). The main result of the paper is that, under appropriate hypotheses on the data, if \((y_{\ast }(\cdot ),u_{\ast }(\cdot ),v_{\ast }(\cdot ))\in _{C}AC_{a+}^{\alpha ,p}\times L_{k}^{\infty }\times L_{k}^{\infty }\) is a local minimum to the above problem, there exists \(\gamma (\cdot )\in I_{b-}^{\alpha }(L_{n}^{p/(p-1)})\), such that \((D_{b-}^{\alpha }\gamma )(s)=\int_{s}^{b}\frac{[\Psi _{y}(t,s,y_{\ast }(s),v_{\ast }(s))]^{T}\gamma (t)}{(t-\tau )^{1-\alpha }}dt+[g_{y}(s,y_{\ast }(s),u_{\ast }(s))]^{T}\gamma (s)+(g_{0})_{y}(s,y_{\ast }(s),u_{\ast }(s),v_{\ast }(s))\), for a.e. \(s\in \lbrack a,b]\), and \(I_{b-}^{1-\alpha }(b)=0\). Here \((D_{b-}^{\alpha }z)(\cdot )\) is the right-sided Riemann-Liouville derivative of order \(\alpha \) of the function \(z\) and \(_{C}AC_{a+}^{\alpha ,p}([a,b]\times \mathbb{R}^{n})\) is the set of all functions \(z(\cdot ):[a,b]\rightarrow \mathbb{R}^{n}\) that have the representation \(z(t)=c_{a}+(I_{a+}^{\alpha }\varphi )(t)\), a.e. \( t\in \lbrack a,b]\), for some \(c_{a}\in \mathbb{R}^{n}\) and \(\varphi (\cdot )\in L_{n}^{p}\). For the proof, the author recalls properties of the left- and right-sided Riemann-Liouville derivatives, and he first proves the optimality conditions in the case of a zero initial condition. The paper ends with the presentation of an example.