Summary
We investigate the existence of periodic solutions to the control problem
with g and f periodic in t with period 1. We form the associated quantities
where (·,·) denotes the inner product inR n and Ω is a nonempty compact set in Rn. If us(t, x), ui(t, x) denote the (in general multivalued) controls for which s(t, x), i(t, x) are respectively attained, then we can form the family of marginal problems
We give sufficient conditions for the existence of a periodic solution of certain marginal problems, stated in terms of\(\mathop {\liminf}\limits_{|x| \to \infty } \) and\(\mathop {\limsup}\limits_{|x| \to \infty } \) of s(t,»)/¦x¦2 and i(t, x)j¦x¦2. Finally we state the relationship between the periodic solutions of the marginal problems and those of the original problem (1).
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The second and third author were partially supported by the research project M.P.I. (40%) «Teoria del controllo dei sistemi dinamici».
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Macki, J.W., Nistri, P. & Zecca, P. Periodic solutions of a control problem via marginal maps. Annali di Matematica pura ed applicata 153, 383–396 (1988). https://doi.org/10.1007/BF01762398
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DOI: https://doi.org/10.1007/BF01762398