In this paper we study a DISCRETE TIME version of deterministic models in optimization in infinite dimensional. The functional are assumed to be merely lower semi continuous. We obtain optimality conditions which are always necessary and which are also sufficient in the convex case whenever the given problem satisfies a qualification condition.

TIME-delays are important components of many dynamical systems that describe coupling or interconnection between dynamics, propagation or transport phenomena, and heredity and competition in population dynamics. The stabilization with TIME delay in observation or control represents difficult mathematical challenges in the control of distributed parameter systems. It is well-known that the stability of closed-loop system achieved by some stabilizing output feedback laws may be destroyed by whatever small TIME delay there exists in observation. In this paper a new method for eigenvalue assignment of DISCRETE-TIME linear systems with state and input TIME-delays by static output feedback matrix is presented. The main result is an iterative method that only requires linear equations to be solved at each iteration. In this scheme, first a linear delayed system by defining an augmented vector is changed to standard form, then output feedback matrix K is calculated by inverse eigenvalue problem. We investigate all types of delays in the states, inputs or both for DISCRETE- TIME linear systems. A simple algorithm and an illustrative example are presented to show the advantages of this new technique.

This paper deals with the fault diagnosis of a linear continuous variable dynamical system represented by a DISCRETE state-space model. The proposed fault diagnoser was based on a special Petri net (PN) called continuous-TIME delay Petri net (CTDPN). According to the theorem presented in this paper, an exact correspondence between DISCRETE state-space equations and fundamental equations of the CTDPN was established. Based on this theorem, a systematic method was proposed for the realization of classical parity equations by a CTDPN that played the role of the fault diagnoser. By integrating the concept of state-space models and PNs in this paper, new and effective methods can be proposed for the analysis and fault diagnosis of hybrid systems. Finally, the performance of the proposed method was investigated for the fault diagnosis of a DC motor. The results demonstrated that, with the help of the proposed PN, fault diagnosis can be performed well and the traditional diagnoser can be replaced with this novel network.