An approach to solve finite time horizon suboptimal feedback control problems for partial differential equations is proposed by solving dynamic programming equations on adaptive sparse grids. A semi-discrete optimal control problem is introduced and the feedback control is derived from the corresponding value function. The value function can be characterized as the solution of an evolutionary HamiltonâJacobi Bellman (HJB) equation which is defined over a state space whose dimension is equal to the dimension of the underlying semi-discrete system. Besides a low dimensional semi-discretization it is important to solve the HJB equation efficiently to address the curse of dimensionality. We propose to apply a semi-Lagrangian scheme using spatially adaptive sparse grids. Sparse grids allow the discretization of the value functions in (higher) space dimensions since the curse of dimensionality of full grid methods arises to a much smaller extent

In this paper semi-smooth Newton methods for optimal control problems governed by the dynamical Lamé system are considered and their convergence behavior with respect to superlinear convergence is analyzed. Techniques from Kröner, Kunisch, Vexler (2011), where semi-smooth Newton methods for optimal control of the classical wave equation are considered, are transferred to control of the dynamical Lamé system. Three different types of control actions are examined: distributed control, Neumann boundary control and Dirichlet boundary control. The problems are discretized by finite elements and numerical examples are presented. 2 U , subject to y = S(u), y ∈ Y, u ∈ U ad ⊂ U ω with control space U ω , state space Y and α > 0. The control-to-state operator S : U ω → Y is assumed to be affine-linear, the functional G : Y → R to be quadratic. The control and state space and the operators are defined in more detail in the next section. The choice of the control-to-state operator incorporates distributed as well as Neumann and Dirichlet boundary control problems of the dynamical Lamé system which we will consider later. The set of admissible controls is defined by U ad = { u ∈ U ω | u a ≤ u ≤ u b } for given u a , u b ∈ U ω. To specify the control-to-state operator we introduce the dynamical Lamé system. Let Ω ⊂ R d , d = 1, 2, 3, be a bounded domain with C 2-boundary (bounded

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Abstract. The feedback stabilization of the Burgers system to a nonstationary solution using finite-dimensional internal controls is considered. Estimates for the dimension of the controller are derived. In the particular case of no constraint in the support of the control a better estimate is derived and the possibility of getting an analogous estimate for the general case is discussed; some numerical examples are presented illustrating the stabilizing effect of the feedback control, and suggesting that the existence of an estimate in the general case analogous to that in the particular one is plausible.MSC2010: 93B52, 93C20, 93D15, 93C50