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Nonlinearly constrained optimal control problems. (English) Zbl 0764.49017
This paper deals with a dynamic object described by \((*)\): \(\dot x(t)=f(t,x(t),u(t))\), \(x(0)=x_ 0\in\mathbb{R}^ n\), where \(x:\mathbb{R}^ 1\to\mathbb{R}^ n\) is the state, \(u:\mathbb{R}^ 1\to\mathbb{R}^ m\) a measurable control function, \(f:\mathbb{R}^ 1\times\mathbb{R}^ n\times\mathbb{R}^ m\to\mathbb{R}^ n\), \(u\in U\equiv\{v=[v_ 1,\dots,v_ m]^ T\in\mathbb{R}^ r:\alpha_ i\leq v_ i\leq\beta_ i\), \(i=1,\dots,r\}\) where \(U\) is a compact and convex set of admissible controls \(u\). Given are the constraints \((**)\): \(\Phi_ i(x(T| u))\leq 0\) \((i=1,\dots,N_ 1)\); \(\Psi_ i(x(T| u))=0\) \((i=1,\dots,N_ E)\); \(g_ i(t,x(t)| u)\) \((t\in[0,T]\), \(i=1,\dots,N)\), where \(\Phi_ i,\Psi_ i\) are real-valued functions on \(\mathbb{R}^ n\), \(g_ i\) are real-valued functions on \([0,T]\times\mathbb{R}^ n\). \({\mathcal F}\) denotes the subset of \(U\) which consists of all controls satisfying \((**)\).
The aim of the work is to find a control function \(u\in{\mathcal F}\) which minimizes the functional \(J(u)=\Phi_ 0(x(T| u))+\int^ T_ 0{\mathcal L}_ 0(t,x(t| u),u(t))dt\) defined over \({\mathcal F}\), where \(\Phi_ 0\), \({\mathcal L}_ 0\) are given real functions. This paper extends the results of former works by the first author as it generalizes a class of constrained optimal control problems.
The approach of finding an optimal solution to the described problem is exemplified on a practical example concerning the transfer of containers from a ship to a cargo truck. The results of this work may be used in different technical problems.
Reviewer: W.Hejmo (Kraków)

49L99 Hamilton-Jacobi theories
49K15 Optimality conditions for problems involving ordinary differential equations
93C15 Control/observation systems governed by ordinary differential equations
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