We investigate the existence of periodic solutions to the control problem $$dot x = f(t,x,u) + g(t), x in R^n , u in R^m ,$$ ((1)) with g and f periodic in t with period 1. We form the associated quantities $$s(t,x) = mathop {sup }limits_{u in Omega } (x,f(t,x,u)), i(t,x) = mathop {inf }limits_{u in Omega } (x,f(t,x,u))$$ where (·,·) denotes the inner product inR n and OHgr 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 $$dot x in lambda (t)overline {co} f(t,x,u_s (t,x)) + (1 - lambda (t))overline {co} f(t,x,u_i (t,x)) + g(t),lambda ( cdot ) in L^infty ([0,1],[0,1]).$$ ((2)) We give sufficient conditions for the existence of a periodic solution of certain marginal problems, stated in terms of $$mathop {lim inf}limits_{|x| o infty } $$ and $$mathop {lim sup}limits_{|x| o infty } $$ 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).
Periodic solutions of a control problem via marginal maps / J. Macki; P. Nistri; P. Zecca. - In: ANNALI DI MATEMATICA PURA ED APPLICATA. - ISSN 0373-3114. - STAMPA. - 153 n.1:(1988), pp. 383-396. [10.1007/BF01762398]
Periodic solutions of a control problem via marginal maps
NISTRI, PAOLO;ZECCA, PIETRO
1988
Abstract
We investigate the existence of periodic solutions to the control problem $$dot x = f(t,x,u) + g(t), x in R^n , u in R^m ,$$ ((1)) with g and f periodic in t with period 1. We form the associated quantities $$s(t,x) = mathop {sup }limits_{u in Omega } (x,f(t,x,u)), i(t,x) = mathop {inf }limits_{u in Omega } (x,f(t,x,u))$$ where (·,·) denotes the inner product inR n and OHgr 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 $$dot x in lambda (t)overline {co} f(t,x,u_s (t,x)) + (1 - lambda (t))overline {co} f(t,x,u_i (t,x)) + g(t),lambda ( cdot ) in L^infty ([0,1],[0,1]).$$ ((2)) We give sufficient conditions for the existence of a periodic solution of certain marginal problems, stated in terms of $$mathop {lim inf}limits_{|x| o infty } $$ and $$mathop {lim sup}limits_{|x| o infty } $$ 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).I documenti in FLORE sono protetti da copyright e tutti i diritti sono riservati, salvo diversa indicazione.