A pointwise constrained version of the Liapunov convexity theorem for single integrals

摘要

Given any AC solution ({overline{x} : [a,b] rightarrow mathbb{R}^{n}}) to the convex ordinary differential inclusion $$x^{prime} ( t) in co{v^{1} ( t), ldots, v^{m} ( t)} qquad a.e. on [ a,b], qquad qquad (^{*})$$we aim at solving the associated nonconvex inclusion $$x^{prime} ( t) in {v^{1} ( t), ldots, v^{m} ( t)} qquad a.e.,x( a) = overline{x} ( a), x( b) = overline{x} ( b), qquad qquad (^{**})$$under an extra pointwise constraint (e.g. on the first coordinate): $$x_{1} ( t) leq overline{x}_{1} ( t) qquad forall t in [ a,b]. qquad qquad qquad (^{***})$$While the unconstrained inclusion (**) had been solved already in 1940 by Liapunov, its constrained version, with (***), was solved in 1994 by Amar and Cellina in the scalar n = 1 case. In this paper we add an extra geometrical hypothesis which is necessary and sufficient, in the vector n > 1 case, for it existence of solution to the constrained inclusion (**) and (***). We also present many examples and counterexamples to the 2 × 2 case. Mathematics Subject Classification (2010) Primary 28B05 Secondary 34A60 Keywords Liapunov convexity theorem for single integrals Nonconvex differential inclusions Pointwise constraints Convexity of the range of vector measures The research leading to this paper was performed at: Cima-ue (Math Research Center of Universidade de Évora, Portugal) with financial support from the research project PEst-OE/MAT/UI0117/2011, FCT (Fundação para a Ciência e a Tecnologia, Portugal); and its resulting applications have been presented by A. Ornelas at the International Workshop “Nonlinear differential equations and control”, Milan September 2011.