Dynamic-Programming

 dp_optimal_control Function Solves discrete-time optimal control problem using Dynamic Programming
    %<br>
    % Inputs:
    %   f      - System dynamics function: x(k+1) = f(x(k), u(k), k)
    %   V      - Stage cost function: V(x(k), u(k), k)
    %   S      - Terminal cost function: S(x(kf), kf)
    %   x0     - Initial state
    %   kf     - Final time step
    %   x_grid - State space discretization
    %   u_grid - Control space discretization
    %
    % Outputs:
    %   x_opt  - Optimal state trajectory
    %   u_opt  - Optimal control sequence
    %   J_opt  - Optimal cost-to-go at initial state
    


% Define system dynamics
f = @(x, u, k) 4*x - 6*u;

% Define stage cost (note: includes the 1/2 factor)
V = @(x, u, k) 0.5*(2*x^2 + 4*u^2)

% Define terminal cost
S = @(x, kf) (x - x_target)^2;

% Solve the problem
[x_opt, u_opt, J_opt] = dp_optimal_control(f, V, S, x0, kf, x_grid, u_grid);