Optimal Control Problems with Convex Control Constraints
We investigate optimal control problems with vector-valued controls. As model problem serve the optimal distributed control of the instationary Navier-Stokes equations. We study pointwise convex control constraints, which is a constraint of the form u(x,
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ptimal Control Problems with Convex Control Constraints Daniel Wachsmuth Abstract. We investigate optimal control problems with vector-valued controls. As model problem serve the optimal distributed control of the instationary Navier-Stokes equations. We study pointwise convex control constraints, which is a constraint of the form u(x, t) ∈ U (x, t) that has to hold on the domain Q. Here, U is an set-valued mapping that is assumed to be measurable with convex and closed images. We establish first-order necessary as well as second-order sufficient optimality conditions. And we prove regularity results for locally optimal controls. Mathematics Subject Classification (2000). Primary: 49M05,26E25; Secondary: 49K20. Keywords. Optimal control, convex control constraints, set-valued mappings, regularity of optimal controls, second-order sufficient optimality condition, Navier-Stokes equations.
1. Introduction In fluid dynamics the control can be brought into the system by blowing or suction on the boundary. Then the control is a velocity, which is a directed quantity, hence it is a vector in R2 respectively R3 . That is, the optimal control problem is to find a vector-valued function u ∈ Lp ((0, T ) × Ω)n . Distributed control can be realized for instance as a force induced by an outer magnet field in a conducting fluid, see, e.g., Kunisch and Griesse [14]. There, the control u is a function of class L2 (Q)2 = L2 (Q; R2 ). This illustrates that the control is a directed quantity: it consists of a direction and an absolute value. Or in other words, the control u at a point (x, t) is a vector in R2 . The optimization has to take into account that one is not able to realize arbitrarily large controls. To this end, control constraints are introduced. If the control u(x, t) is only a scalar variable such as heating or cooling then there is only
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one choice of a convex pointwise control constraint: the so-called box constraints ua (x, t) ≤ u(x, t) ≤ ub (x, t).
(1a)
For the analysis of optimal control of non-stationary Navier-Stokes equations using this particular type of control constraints, we refer to Hinze and Hinterm¨ uller [16], Roub´ıˇccek and Tr¨ oltzsch [23], Tr¨ oltzsch and Wachsmuth [25], and Wachsmuth [27]. But these box constraints are not the only choice for vector-valued controls. For instance, if one wants to bound the R2 -norm of the control, one gets a nonlinear constraint |u(x, t)| = u1 (x, t)2 + u2 (x, t)2 ≤ ρ(x, t). (1b) What happens if the control is not allowed to act in all possible directions but only in directions of a segment with an angle less than π? Using polar coordinates ur (x, t) and uφ (x, t) for the control vector u(x, t), this can be formulated as 0 ≤ ur (x, t) ≤ ψ(uφ (x, t), x, t),
(1c)
where the function ψ models the shape of the set of allowed control actions. Here, we will use another – and more natural – representation of the constraints. Let us denote by U the set of admissible control vectors. Then we can write the control constraints (1a)–(1c) as an inclusion u(x, t) ∈ U
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