Nonlinear Analysis 41 (2000) 631 – 647 www.elsevier.nl/locate/na Multivalued dynamics on a closed domain with absorbing boundary. Applications to optimal control problems with integral constraints M. Motta, F. Rampazzo * Dipartimento di Matematica Pura e Applicata, Universit a di Padova, via Belzoni 7, 35131 Padova, Italy Received 1 April 1998; accepted 13 July 1998 Keywords: Lipschitz dierential inclusions; State constraints; Minimum problems 1. Introduction When a state variable y ∈ R n is subject to a multivalued dynamics ˙ y ∈ F (s; y) (1.1) as, for instance, in the case of control systems, a natural question arises: (Q) given initial values t 1 ;t 2 ¡T; x 1 ;x 2 ∈ R n and a trajectory y 1 (·) on [t 1 ;T ] of the dierential inclusion (1.1) satisfying y 1 (t 1 )= x 1 , can a trajectory y 2 (·) of Eq. (1.1) be found such that y 2 (t 2 )= x 2 and y 2 (·) converges to y 1 (·) (in a suitable sense) as (t 2 ;x 2 ) tends to (t 1 ;x 1 )? In the language of the theory of multifunctions, a positive answer to question (Q) means that the (multivalued) solution map – which to each (t; x) associates all trajec- tories issuing from x at time t – is lower semicontinuous. Some applications motivate the interest for such a property. For example, it guarantees the continuity of the value function of a Bolza problem with a continuous nal cost. When F is measurable in s and Lipschitz in y, an answer to (Q) has been given by means of Filippov’s celebrated theorem. We point out that the Lipschitz condition seems to be crucial, as can be argued by reference to the univalued case. For this very reason an extension of Filippov’s result to the case where the state variable is constrained in the closure  of an open subset  ⊂ R n is not trivial. In fact, if T y  * Corresponding author. Fax: +39-49-875-8596. E-mail address: rampazzo@math.unipd.it (F. Rampazzo) 0362-546X/00/$ - see front matter ? 2000 Elsevier Science Ltd. All rights reserved. PII: S0362-546X(98)00301-0