Isaac Klickstein

Isaac Klickstein

Engineering PhD applying numerical methods to solve controls and optimization problems

Optimal Control Problem: Average Final State

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Average Final State Minimization

\[\begin{aligned} \min && &J = \frac{\alpha}{M} \sum_{i=0}^{M-1} (x_i(t_f) - c_x)^2 + (y_i(t_f) - c_y)^2 + \frac{1}{2} \int_0^{t_f} \left( u_x^2(t) + u_y^2(t) \right) dt\\ \text{s.t.} && &\dot{x}_i(t) = v_i(t), \quad x_i(0) = x_{i,0}\\ && &\dot{v}_i(t) = u_x(t), \quad v_i(0) = v_{i,0}, \quad v_i(t_f) = 0\\ && &\dot{y}_i(t) = w_i(t), \quad y_i(0) = y_{i,0}\\ && &\dot{w}_i(t) = u_y(t), \quad w_i(0) = w_{i,0}, \quad w_i(t_f) = 0\\ && &i = 0,1,\ldots,M-1\\ && &\sum_{i=0}^{M-1} x_i(t_f) = \bar{x}\\ && &\sum_{i=0}^{M-1} y_i(t_f) = \bar{y} \end{aligned}\]

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