Control of Heat Equation with Actuator Placement - Revision history

FelixMueller: /* Parameters */

2016-03-07T10:52:33Z

Parameters

2016-03-07T10:52:01Z

Parameters

2016-03-07T10:51:06Z

Mathematical formulation

← Older revision		Revision as of 10:51, 7 March 2016
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	& u & =0 &\text{ on } &\partial\Omega\times T \\[10pt]		& u & =0 &\text{ on } &\partial\Omega\times T \\[10pt]
	& u(x, y, 0) & = 100 \sin(\pi x)\sin(\pi y) &\text{ in }& \Omega\\[10pt]		& u(x, y, 0) & = 100 \sin(\pi x)\sin(\pi y) &\text{ in }& \Omega\\[10pt]
	& \|v_l\| & \leq M w_l \forall l\in \{1,\dots,L\} &\text{ in } & T \\[10pt]		& \|v_l\| & \leq M w_l \quad \forall l\in \{1,\dots,L\} &\text{ in } & T \\[10pt]
	& \sum\limits_{l=1}^L w_l & \leq W &\text{ in } & T\\[10pt]		& \sum\limits_{l=1}^L w_l & \leq W &\text{ in } & T\\[10pt]
	& w_l(t) & \in \{0,1\} \forall l\in \{1,\dots,L\} &\text{ in } &T.		& w_l(t) & \in \{0,1\} \quad \forall l\in \{1,\dots,L\} &\text{ in } &T.

	\end{array}		\end{array}

2016-03-07T10:50:43Z

Mathematical formulation

← Older revision		Revision as of 10:50, 7 March 2016
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	& u & =0 &\text{ on } &\partial\Omega\times T \\[10pt]		& u & =0 &\text{ on } &\partial\Omega\times T \\[10pt]
	& u(x, y, 0) & = 100 \sin(\pi x)\sin(\pi y) &\text{ in }& \Omega\\[10pt]		& u(x, y, 0) & = 100 \sin(\pi x)\sin(\pi y) &\text{ in }& \Omega\\[10pt]
	& \|v_l\| & \leq M w_l \~~text{ for all }~~ l\in \{1,\dots,L\} &\text{ in } & T \\[10pt]		& \|v_l\| & \leq M w_l \forall l\in \{1,\dots,L\} &\text{ in } & T \\[10pt]
	& \sum\limits_{l=1}^L w_l & \leq W &\text{ in } & T\\[10pt]		& \sum\limits_{l=1}^L w_l & \leq W &\text{ in } & T\\[10pt]
	& w_l(t) & \in \{0,1\} \~~text{ for all }~~ l\in \{1,\dots,L\} &\text{ in } &T.		& w_l(t) & \in \{0,1\} \forall l\in \{1,\dots,L\} &\text{ in } &T.

	\end{array}		\end{array}

2016-03-07T10:49:08Z

Mathematical formulation

← Older revision		Revision as of 10:49, 7 March 2016
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	& u & =0 &\text{ on } &\partial\Omega\times T \\[10pt]		& u & =0 &\text{ on } &\partial\Omega\times T \\[10pt]
	& u(x, y, 0) & = 100 \sin(\pi x)\sin(\pi y) &\text{ in }& \Omega\\[10pt]		& u(x, y, 0) & = 100 \sin(\pi x)\sin(\pi y) &\text{ in }& \Omega\\[10pt]
	& ~~-M w_l~~ & ~~\leq v_l~~\leq M w_l \text{ for all } l\in \{1,\dots,L\} &\text{ in } & T \\[10pt]		& \|v_l\| & \leq M w_l \text{ for all } l\in \{1,\dots,L\} &\text{ in } & T \\[10pt]
	& \sum\limits_{l=1}^L w_l & \leq W &\text{ in } & T\\[10pt]		& \sum\limits_{l=1}^L w_l & \leq W &\text{ in } & T\\[10pt]
	& w_l(t) & \in \{0,1\} \text{ for all } l\in \{1,\dots,L\} &\text{ in } &T.		& w_l(t) & \in \{0,1\} \text{ for all } l\in \{1,\dots,L\} &\text{ in } &T.

2016-03-07T10:47:50Z

Mathematical formulation

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	<p>		<p>
	<math>		<math>
	\begin{array}{~~llcl~~}		\begin{array}{lllcl}

	\min\limits_{u,v,w}~~ &\|\|u(t_f)\|\|_{2,\Omega}^2 +2\|\|u\|\|_{2,\Omega\times T}^2+\frac{1}{500}\sum\limits_{l=1}^L\|\|v_l\|\|^2_{2,T} & \\[10pt]		\min\limits_{u,v,w}~~ &\|\|u(t_f)\|\|_{2,\Omega}^2 &+2\|\|u\|\|_{2,\Omega\times T}^2+\frac{1}{500}\sum\limits_{l=1}^L\|\|v_l\|\|^2_{2,T} & \\[10pt]
	\text{ s.t.} ~~~~ &\frac{\partial u}{\partial t}- \kappa \Delta u=\sum\limits_{l=1}^L v_l f_l &\text{ in }&\Omega\times T\\[10pt]		\text{ s.t.} ~~~~ &\frac{\partial u}{\partial t}- \kappa \Delta u & =\sum\limits_{l=1}^L v_l f_l &\text{ in }&\Omega\times T\\[10pt]
	& u =0 &\text{ on } &\partial\Omega\times T \\[10pt]		& u & =0 &\text{ on } &\partial\Omega\times T \\[10pt]
	& u(x, y, 0) = 100 \sin(\pi x)\sin(\pi y) &\text{ in }& \Omega\\[10pt]		& u(x, y, 0) & = 100 \sin(\pi x)\sin(\pi y) &\text{ in }& \Omega\\[10pt]
	& -M w_l\leq v_l\leq M w_l \text{ for all } l\in \{1,\dots,L\} &\text{ in } & T \\[10pt]		& -M w_l & \leq v_l\leq M w_l \text{ for all } l\in \{1,\dots,L\} &\text{ in } & T \\[10pt]
	& \sum\limits_{l=1}^L w_l \leq W &\text{ in } & T\\[10pt]		& \sum\limits_{l=1}^L w_l & \leq W &\text{ in } & T\\[10pt]
	& w_l(t)\in \{0,1\} \text{ for all } l\in \{1,\dots,L\} &\text{ in } &T.		& w_l(t) & \in \{0,1\} \text{ for all } l\in \{1,\dots,L\} &\text{ in } &T.

	\end{array}		\end{array}

2016-03-07T10:45:32Z

Mathematical formulation

← Older revision		Revision as of 10:45, 7 March 2016
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	\text{ s.t.} ~~~~ &\frac{\partial u}{\partial t}- \kappa \Delta u=\sum\limits_{l=1}^L v_l f_l &\text{ in }&\Omega\times T\\[10pt]		\text{ s.t.} ~~~~ &\frac{\partial u}{\partial t}- \kappa \Delta u=\sum\limits_{l=1}^L v_l f_l &\text{ in }&\Omega\times T\\[10pt]
	& u =0 &\text{ on } &\partial\Omega\times T \\[10pt]		& u =0 &\text{ on } &\partial\Omega\times T \\[10pt]
	& u(0) = 100 \sin(\pi x)\sin(\pi y) &\text{ in }& \Omega\\[10pt]		& u(x, y, 0) = 100 \sin(\pi x)\sin(\pi y) &\text{ in }& \Omega\\[10pt]
	& -M w_l\leq v_l\leq M w_l \text{ for all } l\in \{1,\dots,L\} &\text{ in } & T \\[10pt]		& -M w_l\leq v_l\leq M w_l \text{ for all } l\in \{1,\dots,L\} &\text{ in } & T \\[10pt]
	& \sum\limits_{l=1}^L w_l \leq W &\text{ in } & T\\[10pt]		& \sum\limits_{l=1}^L w_l \leq W &\text{ in } & T\\[10pt]

2016-03-07T10:44:15Z

Discretization

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	This condition is called SOS-<math>W</math> conditon.		This condition is called SOS-<math>W</math> conditon.

	We remark that the number of discretized binary variables does not depend on the space discretization but ~~it depends~~ on the time discretization. ~~Thats~~ why we ~~taged~~ this problem ~~that it contains~~ mesh-~~independend~~ and mesh-~~dependend~~ integer variables.		We remark that the number of discretized binary variables does not depend on the space discretization but does on the time discretization. That is why we tagged this problem as containing mesh-independent and mesh-dependent integer variables.

	==Reference Solution==		==Reference Solution==

2016-03-03T17:52:02Z

← Older revision		Revision as of 17:52, 3 March 2016
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	Additionally, we assume Dirichlet boundary conditions and initial conditions.		Additionally, we assume Dirichlet boundary conditions and initial conditions.
	Originally, the problem formulation was non-convex.		Originally, the problem formulation was non-convex.
	We overcome this issue by substitution of <math>v(t)w_l(t)</math> by <math>v_l(t)</math> and adding the Big ~~<math>~~M~~</math>~~ formulation.		We overcome this issue by substitution of <math>v(t)w_l(t)</math> by <math>v_l(t)</math> and adding the Big M formulation.

2016-03-03T17:40:34Z

Discretization

← Older revision		Revision as of 17:40, 3 March 2016
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	==Discretization==		==Discretization==

	To solve the problem we apply a "first discretize, then optimize" approach an discretize the components of the problem.		To solve the problem we apply a "first discretize, then optimize" approach and discretize the components of the problem.
	For the heat equation, we use a five-point-stencil in space and the implicit euler in time.		For the heat equation, we use a five-point-stencil in space and the implicit euler in time.
	For <math>i=1,\dots, N-1</math>, <math>j=1,\dots, M-1</math>, and <math>k=0,\dots, T_n-1</math>, this yields:		For <math>i=1,\dots, N-1</math>, <math>j=1,\dots, M-1</math>, and <math>k=0,\dots, T_n-1</math>, this yields: