# On some general almost periodic Optimal Control problems: links with periodic problems and necessary conditions

ESAIM: Control, Optimisation and Calculus of Variations (2007)

- Volume: 14, Issue: 3, page 590-603
- ISSN: 1292-8119

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topPennequin, Denis. "On some general almost periodic Optimal Control problems: links with periodic problems and necessary conditions." ESAIM: Control, Optimisation and Calculus of Variations 14.3 (2007): 590-603. <http://eudml.org/doc/90885>.

@article{Pennequin2007,

abstract = {
In this paper, we are concerned with periodic, quasi-periodic (q.p.) and almost periodic (a.p.) Optimal Control problems. After defining these problems and setting them in an abstract setting by using Abstract Harmonic Analysis, we give some structure results of the set of solutions, and study the relations between periodic and a.p. problems. We prove for instance that for an autonomous concave problem, the a.p. problem has a solution if and only if all problems (periodic with fixed or variable period, q.p. or a.p.) have a constant solution. After that, we give some first order necessary conditions (weak Pontryagin) in the space of Harmonic Synthesis and we also give in this space an existence result.
},

author = {Pennequin, Denis},

journal = {ESAIM: Control, Optimisation and Calculus of Variations},

keywords = {Almost Periodic Optimal Control; Periodic Optimal Control; Pontryagin theorem; Almost periodicity on groups; almost periodic optimal control; periodic optimal control; almost periodicity on groups},

language = {eng},

month = {12},

number = {3},

pages = {590-603},

publisher = {EDP Sciences},

title = {On some general almost periodic Optimal Control problems: links with periodic problems and necessary conditions},

url = {http://eudml.org/doc/90885},

volume = {14},

year = {2007},

}

TY - JOUR

AU - Pennequin, Denis

TI - On some general almost periodic Optimal Control problems: links with periodic problems and necessary conditions

JO - ESAIM: Control, Optimisation and Calculus of Variations

DA - 2007/12//

PB - EDP Sciences

VL - 14

IS - 3

SP - 590

EP - 603

AB -
In this paper, we are concerned with periodic, quasi-periodic (q.p.) and almost periodic (a.p.) Optimal Control problems. After defining these problems and setting them in an abstract setting by using Abstract Harmonic Analysis, we give some structure results of the set of solutions, and study the relations between periodic and a.p. problems. We prove for instance that for an autonomous concave problem, the a.p. problem has a solution if and only if all problems (periodic with fixed or variable period, q.p. or a.p.) have a constant solution. After that, we give some first order necessary conditions (weak Pontryagin) in the space of Harmonic Synthesis and we also give in this space an existence result.

LA - eng

KW - Almost Periodic Optimal Control; Periodic Optimal Control; Pontryagin theorem; Almost periodicity on groups; almost periodic optimal control; periodic optimal control; almost periodicity on groups

UR - http://eudml.org/doc/90885

ER -

## References

top- J.-P. Aubin, Optima and Equilibria: an introduction to Nonlinear Analysis. Springer, 2nd Edn. (1988).
- A.S. Besicovitch, Almost Periodic Functions. Cambridge University Press, Cambridge (1932) (and Dover, 1954).
- J. Blot, Le théorème de Markov-Kakutani et la presque-périodicité, Fixed Point Theory and Applications, M. Théra and J.B. Baillon Eds., Pitman Research Notes in Mathematical Series252, Longman, London (1991) 45–56.
- J. Blot, Oscillations presque-périodiques forcées d'équations d'Euler-Lagrange. Bull. Soc. Math. France122 (1994) 285–304.
- J. Blot, Variational Methods for the Almost Periodic Lagrangian Oscillations. Preprint, Cahiers Eco et Maths No. 96.44 (1996).
- J. Blot and D. Pennequin, Spaces of quasi-periodic functions and oscillations in dynamical systems. Acta Appl. Math.65 (2001) 83–113.
- J. Blot and D. Pennequin, Existence and structure results on Almost Periodic solutions of Difference Equations. J. Diff. Equa. Appl.7 (2001) 383–402.
- H. Bohr, Almost Periodic Functions. Julius Springer, Berlin (1933) (Chelsea Publishing Company, N.Y., 1947).
- F. Colonius, Optimal Periodic Control, in Lect. Notes Math.1313, Springer, Berlin (1988).
- C. Corduneanu, Almost Periodic Functions. Chelsea (1989).
- G. Da Prato and A. Ichikawa, Optimal control of linear systems with a.p. inputs. SIAM J. Control Optim.25 (1987) 1007–1019.
- D.G. De Figueiredo, Lectures on the Ekeland Variational Principle with Applications and Detours. Tata Institute of Fundamental Research, Bombay (1989).
- J. Favard, Leçons sur les fonctions presque-périodiques. Gauthiers-Villars, Paris (1933).
- A. Halanay, Optimal Control of Periodic solutions. Rev. Rouman. Mat. Pure Appl.19 (1974) 3–16.
- V.P. Havin and N.K. Nikolski Eds., Commutative Harmonic Analysis II. Springer, Berlin (1991).
- E. Hewitt, K.A. Ross, Abstract Harmonic Analysis I & II. Springer, Berlin, 2nd Edn. (1979) (and 1970).
- F.J.M. Horn and J.E. Bailey, An application of the theorem of relaxed control to the problem of increasing catalyst selectivity. J. Opt. Theory Appl.2 (1968) 441–449.
- A. Kovaleva, Optimal Control of Mechanical Oscillations. Springer, Berlin (1999).
- J.L. Mauclaire, Intégration et Théorie des Nombres. Travaux en Cours, Hermann, Paris (1986).
- G.M. N'Guérékata, Almost automorphic and almost periodic functions in abstract spaces. Kluwer Academic Publishers (2001)
- P. Nistri, Periodic Control Problems for a class of nonlinear periodic differential systems. Nonlinear Anal. Theor. Meth. Appl.7 (1983) 79–90.
- D. Pennequin, Existence results on almost periodic solutions of discrete time equations. Discrete Cont. Dynam. Syst.7 (2001) 51–60.
- I.C. Percival, Variational principles for the invariant toroids of classical dynamics. J. Phys. A: Math. Nucl. Gen.7 (1974) 794–802.
- I.C. Percival, Variational principles for invariant tori and cantori. A.I.P. Conf. Proc.57 (1979) 302–310.
- L. Pontryagin, Topological Groups. N.Y. Gordon and Breach (1966).
- J.L. Speyer, Periodic optimal flight. J. Guid. Control Dynam.61 (1996) 745–754.
- W. Rudin, Fourier Analysis on Groups. Interscience Publishers, N.Y. (1962).
- A. Weil, L'intégration dans les Groupes Topologiques. Hermann, Paris (1940).

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