Application of the hypodifferential descent method to the problem of constructing an optimal control

Exact penalty functions for optimal control problems I: Main theorem and free‐endpoint problems

“…The reason to suppose q <+ ∞ in this case is related to the analysis of numerical methods. Although “discretise‐then‐optimise”‐type methods are prevalent, there exist some continuous methods for solving optimal control problems (see, eg, References), some of which are based on the minimisation of the penalty function Φ λ ( x , u ). These methods are usually formulated and analysed in the case p = q =2, ie, in the Hilbert space setting, when one can utilise inner products.…”

“…15,16 The main results of these papers were further extended to isoperimetric problems of the calculus of variations, 17 variational problems involving higher order derivatives, 18 parametric moving boundary variational problems, 19 control problems involving differential inclusions, 20 and certain optimal control problems for implicit control systems with strict inequality constraints. 21 Numerical methods for solving optimal control problems based on the use of exact penalty functions in the infinite dimensional setting were probably first considered by Outrata 22 (see also the works of Outrata and Schindler 23,24 ), and later on were also studied in the work of Fominyh et al 25 However, in the work of Outrata, 22 only the local exactness of a penalty function was considered under the assumption that an abstract constraint qualification holds true, and it is unclear how to verify this assumption for any particular problem. In the works of Demyanov et al 11 and Fominyh et al, 25 the global exactness of penalty functions was stated without proof.…”

“…21 Numerical methods for solving optimal control problems based on the use of exact penalty functions in the infinite dimensional setting were probably first considered by Outrata 22 (see also the works of Outrata and Schindler 23,24 ), and later on were also studied in the work of Fominyh et al 25 However, in the work of Outrata, 22 only the local exactness of a penalty function was considered under the assumption that an abstract constraint qualification holds true, and it is unclear how to verify this assumption for any particular problem. In the works of Demyanov et al 11 and Fominyh et al, 25 the global exactness of penalty functions was stated without proof. The main results on exact penalty functions for various variational problems from other works [13][14][15][16][17][18][19] are based on the assumptions that the objective function is Lipschitz continuous on a rather complicated and possibly unbounded set, and a penalty function attains a global minimum in the space of piecewise continuous functions for any sufficiently large value of the penalty parameter, and it is, once again, unclear how to verify these assumptions in any particular case.…”

Exact penalty functions for optimal control problems II: Exact penalization of terminal and pointwise state constraints

“…An exact penalty method for optimal control problems with delay was proposed in [63], and such method for some nonsmooth optimal control problems was studied in the works of Outrata et al [53][54][55]. A continuous numerical method for optimal control problems based on the direct minimisation of an exact penalty function was considered in the recent paper [29]. Finally, closely related methods based on Huyer and Neumaier's exact penalty function [23,34,62] were developed for optimal control problems with state inequality constraints [38,44] and optimal feedback control problems [45].…”

“…An exact penalty method for optimal control problems with delay was proposed by Wong and Teo, 23 and such method for some nonsmooth optimal control problems was studied in the works of Outrata et al [24][25][26] A continuous numerical method for optimal control problems based on the direct minimization of an exact penalty function was considered in recent paper. 27 Finally, closely related methods based on Huyer and Neumaier's exact penalty function 11,28,29 were developed for optimal control problems with state inequality constraints 30,31 and optimal feedback control problems. 32 Despite the abundance of publications on exact penalty methods for optimal control problems, relatively little attention has been paid to an actual analysis of the exactness of penalty functions for such problems.…”