Study On Controllability Of Impulsive Hybrid Dynamical Systems

With the rapid development of science and technology,system science is evolving towards the networked,intelligent and complex direction.It is usually difficult to analyze the complex dynamic processes in the real world by using the pure continuous or discrete system theory.Therefore,it is of great theoretical and practical significance to study the hybrid dynamic system,which is formed by the interaction of continuous system and discrete system.In particular,impulsive systems and switched systems,two kinds of important hybrid systems,have been widely used in economic field,mechanical engineering,secure communication,power network,traffic system,energy system and other practical scenarios,and hence have received widespread attention from researchers.The existing controllability theory of hybrid system does not fully consider the characteristics of hybridity,nonlinearity,time delay and time-varying parameter.In view of this,the controllability of several kinds of time-varying impulsive hybrid dynamic systems is studied in this dissertation.The main contents are as follows.For the linear time-varying complex-valued impulsive systems with piecewise delays,the null controllability and complete controllability conditions are established in the form of Gramian type and Kalman type.By constructing special control functions at some impulsive intervals or impulsive instants and applying the relevant theory of linear algebra,the sufficient null controllability conditions under the time-varying case are established.Under the assumption that all impulsive gain matrices are non-degenerate,the necessary conditions for the null controllability of the system are proved.Several sufficient conditions for complete controllability of the system are developed by constructing more general types of control functions and combining the concept of the complete controllability.By constructing the switched control function,using the properties of Hermitian semi-positive definite matrix,the necessary and sufficient conditions for the complete controllability of the system under time-varying situation are established,and then the complete controllability of the time-invariant system is further discussed.Numerical examples are given to verify the controllability criteria for several different systems.For the linear time-varying impulsive and switching systems with time delays,some sufficient conditions and necessary and sufficient conditions for the complete controllability are established by designing the discontinuous control functions.Firstly,based on the concept of state transition matrix,by using constant variation method and mathematical induction method,and combining the integral variable substitution formula,the general solution expression of the system on every resident interval is developed.Then,on the premise that the partial impulsive gain matrices are non-degenerate,several sufficient criteria for the complete controllability are established by using the constant variation formula of the system solution and constructing a discontinuous explicit controller.Without assuming that the impulsive matrix is non-degenerate,the necessary and sufficient conditions for the complete controllability of the system are developed by using the properties of real symmetric semidefinite matrices and the theory of characteristic polynomials.For the case of time-invariant system,by using the geometric theory of linear system,the rank condition of Gramian matrix is equivalently converted to the easy-verified rank condition of Kalman type.Finally,two numerical simulations of time-varying system and time-invariant system are used to verify the theoretical results.For the discrete-time impulsive hybrid systems with input delay,the invariant subspace sequence related to the system matrices is constructed,and several criteria of null reachability,null controllability and complete controllability are established.Firstly,the null reachable set and null controllable set of the system with respect to an impulsive and switching sequence are established by using the geometric method and the properties of invariant subspace.The necessary and sufficient conditions of null reachability and null controllability of the system are proved by constructing the subspace sequence and analyzing the relationship between adjacent subspaces.Then,the state transition matrix of time-varying discrete systems is introduced,and some sufficient rank conditions for complete controllability of the system under time-varying case are established by the algebraic method.Combined with matrix theory and other related theories,a less conservative necessary and sufficient condition for the complete controllability is further obtained.Finally,two numerical examples are used to verify the validity of the theoretical results.For a class of piecewise nonlinear impulsive non-autonomous systems,the control mapping and nonlinear operator dependent on the controllability condition of piecewise linear part are constructed,and the system controllability criteria under several kinds nonlinear constraints are established.A control mapping and a switched nonlinear operator are constructed in a Banach space composed of all piecewise continuous functions on a closed interval,and then the controllability problem is transformed into the existence problem of fixed point of nonlinear operator.When the nonlinearity satisfies the linear growth condition,the existence of the fixed point of the nonlinear operator is proved and the sufficient condition of the controllability of the system is established by inequality condition and Schauder’s fixed point theorem.When the nonlinearity satisfies the bounded constraint,this kind of constraint condition is transformed into a special case of the linear growth condition,and a more concise condition of controllability of the system is established by using Schauder’s fixed point theorem.When the nonlinearity satisfies the sublinear growth condition,the continuity and relative compactness of the nonlinear operator on a bounded closed convex set are proved and then another controllability criterion of the system is established by the matrix inequality and Rothe’s fixed point theorem.Numerical examples confirm the controllability theory under different nonlinear constraints.For the nonlinear non-autonomous impulsive and switching systems with time delay,the controllability conditions under Lipschitz and linear growth nonlinear constraints are derived respectively.Firstly,on the premise that some input matrix of linear part have generalized inverse,a piecewise continuous control mapping is constructed.Based on the control mapping and the equivalent integral form of the system,a piecewise nonlinear operator is constructed on the piecewise continuous function space.The sufficient condition of system controllability under Lipschitz constraint is established by Banach compression mapping principle.Then,on the premise of the controllability of the linear time-delay switching part,a switching control mapping and nonlinear operator that rely on the controllability matrix of the linear part are constructed,and another sufficient condition for system controllability is established by using matrix inequality theory and Rothe’s fixed point theorem.Finally,numerical examples are used to verify the established controllability results.