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Mixed constraints in optimal control: An implicit function theorem approach. (English) Zbl 1124.49013

As the authors state in the Introduction, “our aim is to give a clear explanation of how the technique used by [M. Hestenes, Calculus of variations and optimal control, New York-London-Sydney: J. Wiley & Sons (1966; Zbl 0173.35703)] to obtain second-order necessary conditions for unconstrained problems can be used to derive such conditions for problems involving mixed, state-control equality and/or inequality constraints, once the original problem is reduced to a problem without constraints”.
More precisely, the authors consider a solution, \((x_0(.),u_0(.))\), of the optimal control problem of minimizing the functional \[ I(x(.),u(.)):=\int_{t_0}^{t_1}L(t,x(t),u(t))\,dt \] subject to: \[ x'(t)=f(t,x(t),u(t)), \;t\in T:=[t_0,t_1], \;x(t_0)=\xi_0, \;x(t_1)=\xi_1, \]
\[ (t,x(t),u(t))\in{\mathcal A}\subseteq T\times R^n\times R^m \;\forall t\in T \] in the class of piecewise continuous admissible controls, \(u(.)\), and look for second-order necessary conditions.
Resuming, simplifying and systematizing some of the arguments in M. Hestenes’ book, the authors obtain the same type of second-order necessary conditions for “normal extremals”, in three different cases: the un-constrained case in which \({\mathcal A}\subseteq T\times R^n\times R^m\) is relatively open, the case of equality constraints in which \({\mathcal A}:=\{(t,x,u)\); \( \varphi (t,x,u)=0\}\) and the mixed-constraints case in which \({\mathcal A}:=\{(t,x,u)\); \( \varphi_\alpha (t,x,u)=0\), \(\alpha =1,2,\dots, r\), \(\varphi_\beta (t,x,u)=0\), \(\beta =r+1,\dots, q\}\) which satisfy suitable “constraint qualifications”; the main idea is to use one of the Hestenes’ implicit functions theorems to reduce the last two cases to the first one.

MSC:

49K15 Optimality conditions for problems involving ordinary differential equations
49K27 Optimality conditions for problems in abstract spaces

Citations:

Zbl 0173.35703
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