# An SQP-type Solution Method for Constrained Discrete-Time Optimal Control Problems

• E. Arnold
• H. Puta
Chapter
Part of the ISNM International Series of Numerical Mathematics book series (ISNM, volume 115)

## Abstract

The considered nonlinear, constrained discrete-time optimal control problem is stated as follows:
$$J = F({x^N}) + \sum\limits_{k = 0}^{N -1} {f_0^k} ({x^k},{u^k}) \to Min$$
subject to the state equation:
$${x^{k + 1}} = {f^k}({x^k},{u^k}),k = 0, \ldots ,N -1,$$
(1)
and inequality constraints:
$$\begin{gathered} {{c}^{k}}({{x}^{k}},{{u}^{k}}) \leqslant 0,\quad k = 0, \ldots ,N - 1,{\kern 1pt} \hfill \\ {\kern 1pt} {{c}^{N}}({{x}^{N}}) \leqslant 0, \hfill \\ {{f}^{k}}:{{{\text{R}}}^{n}} \times {{{\text{R}}}^{m}} \to {{{\text{R}}}^{n}},{{c}^{k}}:{{{\text{R}}}^{n}} \times {{{\text{R}}}^{m}} \to {{{\text{R}}}^{{{{r}^{k}}}}} \hfill \\ \end{gathered}$$
with sufficiently smooth functions F, ƒ 0 k , ƒ k , c k . The constraints include fixed initial or final states as well as bounds for state and control variables or more general constraints.

## Keywords

Constrained optimal control discrete-time systems structured nonlinear programming hydroelectric power-station systems

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