Tham khảo tài liệu 'optimal control with engineering applications episode 2', kỹ thuật - công nghệ, cơ khí - chế tạo máy phục vụ nhu cầu học tập, nghiên cứu và làm việc hiệu quả | 2 List of Symbols Integers i j k I m indices dimension of the control vector n dimension of the state and the costate vector p X0 dimension of an output vector scalar Lagrange multiplier for J 1 in the regular case 0 in a singular case Functions f . f x u t g G . h . 9 . function in a static optimization problem right-hand side of the state differential equation define equality or inequality side-constraints switching function for the control and offset function in a singular optimal control problem H x u X Xo t J u J x t L x u t K x tb A t B t C t D t F Q t R t N t G t K t Hamiltonian function cost functional optimal cost-to-go function integrand of the cost functional final state penalty term system matrices of a linear time-varying system penalty matrices in a quadratic cost functional state-feedback gain matrix solution of the matrix Riccati differential equation in an LQ regulator problem P t Q t R t S t K . observer gain matrix noise intensity matrices in a stochastic system state error covariance matrix support function of a set Operators d dt E . . t T U df dx total derivative with respect to the time t expectation operator taking the transpose of a matrix adding a matrix to its transpose Jacobi matrix of the vector function f with respect to the vector argument x VxL gradient of the scalar function L with respect to x T fdL T xL ax 1 Introduction Problem Statements In this book we consider two kinds of dynamic optimization problems optimal control problems and differential game problems. In an optimal control problem for a dynamic system the task is finding an admissible control trajectory u ta tb Q c Rm generating the corresponding state trajectory x ta tb Rn such that the cost functional J u is minimized. In a zero-sum differential game problem one player chooses the admissible control trajectory u ta tb Qu c Rmu and another player chooses the admissible control trajectory v ta tb Qv c Rmv. These choices generate the corresponding state trajectory x
