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I am trying to solve a control problem with control $u$ and state $x$ with the following structure: $$\max_{u(\cdot),x(\cdot)} \int_0^1 f(x(t),u(t),t)dt,$$ subject to $$x'(t)=g(x(t),u(t),t),$$ $$\int_0^1 x(t)dt \geq x(t_o)+ K,$$ where $t_o\in(0,1)$ and $K>0$ are exogenously given parameters. Note the big difference here with other control problems is that the value of the state at a given point $t=t_o$, $x(t_o)$, is part of the integral constraint.

Would be very appreciative if someone could point me in a general direction about where to find a recipe for this sort of control problem. Thanks in advance!

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The general approach for dealing with such a problem is to use of Lagrange multiplier and define a scalar $\lambda>0$ and consider the following objective function to maximize

$$\int_0^1f(x(t),u(t),t)dt+\lambda\left(\int_0^1(x(t)-x(t_o)-K)dt\right)$$

and apply the usual methods for solving it, that is, either using dynamic programming or the maximum principle. For solving for the optimal control law, you will need to consider the complemetary slackness condition that $$\lambda^*\left(\int_0^1(x^*(t)-x(t_o)-K)dt\right)=0$$ where the $*$ superscript denotes optimality. This may be difficult to analytically verify at it will require the exact computation of $x^*(t_0)$, which may not be possible.

Note however, that this constraint may be considered in numerical solvers more easily.

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