A Class Of Time Optimal Control Problems Dominated By Impulsive Differential Equations

Pulse is a very common phenomenon in nature(pulse phenomenon or instantaneous disturbance phenomenon for short),and also a technique widely used in engineering(pulse control technique),such as supernova explosions(internal fusion of stars),the outbreak of plague,epidemics,the orbit transfer during satellite launch and so on.On the other hand,the impulsive differential equation is a powerful tool used to describe dynamic systems with impulsive phenomena.Menwhile,the optimal time control problem has been the core of optimal control problem research since the birth of cybernetics.Though integral optimal control problems have achieved fruitful results,whether theories or numerical calculations.However,since the solution of optimal time control is a bound of a set,up to now,a better result of the numerical calculation of optimal time control has not been developed,which is a well-known puzzle.To tackle this problem,it is very beneficial to explore a certain equivalence between the optimal time control problem and a class of integral-type optimal control problem.Therefore,it is undoubtedly an important subject to study the optimal time control problem dominated by the impulse system.In this paper,our purpose is to investigate the optimal time control problem with impulsive linear ordinary differential equation and impulsive heat equation,and to reveal the essential similarities and differences between impulsive differential equation and differential equation.To this end,we first further discuss the optimal time control problem dominated by the impulse linear ordinary differential equation,and find that the impulse will destroy the controllability of the original system.Consequently,the impulse conditions is introduced reasonably to discuss the controllability of the impulse to the system.Secondly,the optimal control problem is proposed so as to explore the maximum principle and the bang-bang principle.Then,the equivalence relations between the optimal time control problem,the optimal norm control problem and the optimal control problem are obtained.On this basis,we discuss the optimal time control problem dominated by the impulse heat equation.In the first place,the approximate and null controllability of the impulse system are achieved by introducing the impulse conditions reasonably.Afterward,the maximum principle and the bang-bang principle of the optimal control problem of the impulse heat equation are given.Finally,the equivalence relations between the optimal time control problem,the optimal norm control problem and the optimal control problem are developed.These results not only set up a actionable base for numerical calculation of the optimal time control,but also enrich the research achievements of the optimal time control problem.Moreover,we also find that the pulse can destroy the basic properties of the original system,for example controllability,while discovering the influence of the pulse to default properties of the differential system at the angle of the optimal time control problem.It also reveal that the essential differences between the integral optimal control problem and the optimal time control problem dominated by the impulsive differential equation.