Pontryagin's Minimum Principle - Formal Statement of Necessary Conditions For Minimization Problem

Formal Statement of Necessary Conditions For Minimization Problem

Here the necessary conditions are shown for minimization of a functional. Take to be the state of the dynamical system with input, such that

\dot{x}=f(x,u), \quad x(0)=x_0, \quad u(t) \in \mathcal{U}, \quad t \in

where is the set of admissible controls and is the terminal (i.e., final) time of the system. The control must be chosen for all to minimize the objective functional which is defined by the application and can be abstracted as

J=\Psi(x(T))+\int^T_0 L(x(t),u(t)) \,dt

The constraints on the system dynamics can be adjoined to the Lagrangian by introducing time-varying Lagrange multiplier vector, whose elements are called the costates of the system. This motivates the construction of the Hamiltonian defined for all by:

H(\lambda(t),x(t),u(t),t)=\lambda'(t)f(x(t),u(t))+L(x(t),u(t)) \,

where is the transpose of .

Pontryagin's minimum principle states that the optimal state trajectory, optimal control, and corresponding Lagrange multiplier vector must minimize the Hamiltonian so that

(1) \qquad H(x^*(t),u^*(t),\lambda^*(t),t)\leq H(x^*(t),u,\lambda^*(t),t) \,

for all time and for all permissible control inputs . It must also be the case that

(2) \qquad \Psi_T(x(T))+H(T)=0 \,

Additionally, the costate equations

(3) \qquad -\dot{\lambda}'(t)=H_x(x^*(t),u^*(t),\lambda(t),t)=\lambda'(t)f_x(x^*(t),u^*(t))+L_x(x^*(t),u^*(t))

must be satisfied. If the final state is not fixed (i.e., its differential variation is not zero), it must also be that the terminal costates are such that

(4) \qquad \lambda'(T)=\Psi_x(x(T)) \,

These four conditions in (1)-(4) are the necessary conditions for an optimal control. Note that (4) only applies when is free. If it is fixed, then this condition is not necessary for an optimum.

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