Optimal control, criteria (INPROGRESS)

Cheatsheet for Zadorozhnii' optimal control course. So the credit for whatever's correct goes to Zadorozhnii and all the errors are to be considered mine. Some refernces (specific to the course mentioned):

1. J. Warga: https://libgen.pw/item/detail/id/1199754?id=1199754

Consider a general problem $J(x, u, t_0, t_1) = \int_{t_0}^{t_1} f_0(t, x(t), u(t) \mathrm{d}t + \Phi_0(t_0, t_1, x(t_0), x(t_1)) \longrightarrow \inf,$ subject to $\dot x(t) = f(t, x(t), u(t)),$ $\Phi(t_0, t_1, x(t_0), x(t_1)) = 0.$ $u\in\mathrm{PC}(\mathbb{R}, \mathbb{R}^n).$

Lagrangian

As usual we use the Lagrange's multiplier method in order to reduce the bounded problem to an unbounded one: $L(u, \lambda_0, \lambda, \mu) = \lambda_0 J(x, u) + \int_{t_0}^{t_1} \langle \lambda(t), \left[ f(t, x(t), u(t)) - \dot x(t)\right] \rangle \mathrm{d}t + \langle \mu, \Phi(t_0, t_1, x(t_0), x(t_1)).$

Hamiltonian

Let us introduce the helper functions with the formulas $H(t, x(t), u(t), \lambda_0, \lambda(t), \mu) = \lambda_0 f_0(t, x(t), u(t)) + \langle \lambda(t), f(t) \rangle,$ $l(t_0, t_1, x(t_0), x(t_1), \mu) = \langle \mu, \Phi(t_0, t_1, x(t_0), x(t_1)) \rangle.$ Now we can rewrite the Lagrangian: $L(\cdots) = \int_{t_0}^{t_1} \left[ H(\cdots) + \langle \dot \lambda(t), x(t) \rangle \right] \mathrm{d}t + l(\cdots) + \langle \lambda(t_0), x(t_0) \rangle - \langle \lambda(t_1), x(t_1) \rangle.$ Here $\langle \lambda(t_0), x(t_0) \rangle - \langle \lambda(t_1), x(t_1) \rangle = \int_{t_0}^{t_1} \langle \lambda(t), -\dot x(t)\rangle \mathrm{d}t.$