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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)

##### MSC:
 49L99 Hamilton-Jacobi theories 49K15 Optimality conditions for problems involving ordinary differential equations 93C15 Control/observation systems governed by ordinary differential equations
##### Keywords:
optimal problem in functional space
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