On the turnpike property with interior decay for optimal control problems
Submission Status: | accepted for publication |
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- In this paper the turnpike phenomenon is studied for problems of optimal control where both pointwise-in-time state and control constraints can appear. We assume that in the objective function, a tracking term appears that is given as an integral over the time-interval [0, T] and measures the distance to a desired stationary state. In the
optimal control problem, both the initial and the desired terminal state are prescribed. We assume that the system is exactly controllable in an abstract sense if the time horizon is long enough.
We show that that the corresponding optimal control problems on the time intervals [0, T] give rise to a turnpike structure in the sense that for natural numbers n if T is su� ciently large, the contribution
of the objective function from subintervals of [0, T] of the form
[t - t/2^n, t + (T-t)/2^n]
is of the order 1/min{t^n, (T-t)^n}. We also show that a similar result holds for epsilon-optimal solutions of the optimal control problems if epsilon > 0 is chosen suffciently small. At the end of the paper we present
both systems that are governed by ordinary differential equations and
systems governed by partial differential equations where the results can be applied.