# A Direct Method for Parabolic PDE Constrained Optimization ProblemsNonlinear boundary control of the periodic 1D heat equation

A Direct Method for Parabolic PDE Constrained Optimization Problems: Nonlinear boundary control... 13 Nonlinear boundary control of the periodic 1D heat equation In this chapter we consider the problem of optimal nonlinear boundary control of the periodic heat equation 1 γ 2 2 minimize (u(1;.) − u ˆ) + (q − q ˆ) (13.1a) 2 2 q∈L (Σ),u∈W(0,1) Ω Σ s. t. ∂ u = DΔu, in (0, 1) ×Ω, (13.1b) 4 4 ∂ u + αu = β q , in (0, 1) × ∂Ω, (13.1c) u(0;.)= u(1;.), in Ω, (13.1d) on Ω =(0, 1). We see that problem (13.1) is very similar to the model prob- lem (6.1) except for the polynomial terms in the boundary control condition (13.1c) of Stefan–Boltzmann type. 13.1 Problem and algorithmical parameters For our computations the desired state and control proﬁles are u ˆ(x)= 1 + cos(π(x − 1))/10, q ˆ(t, x)= 1. The other problem parameters are given by −4 γ = 10 , D = 1, α(t, 0)= β(t, 0)= 1, α(t, 1)= β(t, 1)= 0, effectively resulting in a homogeneous Neumann boundary condition without con- trol at x = 1. The control acts only via the boundary at x = 0. −5 We performed all computations with a relative integrator tolerance http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png

# A Direct Method for Parabolic PDE Constrained Optimization ProblemsNonlinear boundary control of the periodic 1D heat equation

Part of the Advances in Numerical Mathematics Book Series
8 pages

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Publisher
ISBN
978-3-658-04475-6
Pages
171 –179
DOI
10.1007/978-3-658-04476-3_13
Publisher site
See Chapter on Publisher Site

### Abstract

13 Nonlinear boundary control of the periodic 1D heat equation In this chapter we consider the problem of optimal nonlinear boundary control of the periodic heat equation 1 γ 2 2 minimize (u(1;.) − u ˆ) + (q − q ˆ) (13.1a) 2 2 q∈L (Σ),u∈W(0,1) Ω Σ s. t. ∂ u = DΔu, in (0, 1) ×Ω, (13.1b) 4 4 ∂ u + αu = β q , in (0, 1) × ∂Ω, (13.1c) u(0;.)= u(1;.), in Ω, (13.1d) on Ω =(0, 1). We see that problem (13.1) is very similar to the model prob- lem (6.1) except for the polynomial terms in the boundary control condition (13.1c) of Stefan–Boltzmann type. 13.1 Problem and algorithmical parameters For our computations the desired state and control proﬁles are u ˆ(x)= 1 + cos(π(x − 1))/10, q ˆ(t, x)= 1. The other problem parameters are given by −4 γ = 10 , D = 1, α(t, 0)= β(t, 0)= 1, α(t, 1)= β(t, 1)= 0, effectively resulting in a homogeneous Neumann boundary condition without con- trol at x = 1. The control acts only via the boundary at x = 0. −5 We performed all computations with a relative integrator tolerance

Published: Nov 30, 2013

Keywords: Coarse Grid; Numerical Convergence; Cumulative Time; Mesh Level; Hessian Approximation