543 research outputs found
Abstract nonlinear sensitivity and turnpike analysis and an application to semilinear parabolic PDEs
We analyze the sensitivity of the extremal equations that arise from the
first order necessary optimality conditions of nonlinear optimal control
problems with respect to perturbations of the dynamics and of the initial data.
To this end, we present an abstract implicit function approach with scaled
spaces. We will apply this abstract approach to problems governed by semilinear
PDEs. In that context, we prove an exponential turnpike result and show that
perturbations of the extremal equation's dynamics, e.g., discretization errors
decay exponentially in time. The latter can be used for very efficient
discretization schemes in a Model Predictive Controller, where only a part of
the solution needs to be computed accurately. We showcase the theoretical
results by means of two examples with a nonlinear heat equation on a
two-dimensional domain.Comment: 29 pages, 4 figure
Time adaptivity in model predictive control
The core of the Model Predictive Control (MPC) method in every step of the
algorithm consists in solving a time-dependent optimization problem on the
prediction horizon of the MPC algorithm, and then to apply a portion of the
optimal control over the application horizon to obtain the new state. To solve
this problem efficiently, we propose a time-adaptive residual a-posteriori
error control concept based on the optimality system of this optimal control
problem. This approach not only delivers a tailored time discretization of the
the prediction horizon, but also suggests a tailored length of the application
horizon for the current MPC step. We apply this concept for systems governed by
linear parabolic PDEs and present several numerical examples which demonstrate
the performance and the robustness of our adaptive MPC control concept
Differential-Algebraic Equations
Differential-Algebraic Equations (DAE) are today an independent field of research, which is gaining in importance and becoming of increasing interest for applications and mathematics itself. This workshop has drawn the balance after about 25 years investigations of DAEs and the research aims of the future were intensively discussed
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