On dual dynamic programming in shape optimization of coupled models
Abstract
We propose a new method for analysis of shape optimization of coupled models. The framework of the dual dynamic programming is introduced for a solution of the problems. The shape optimization of coupled model is formulated in terms of characteristic functions which define the suport of control. The construction of εoptimal solution of such a problem can be obtained by solving the sufficient εoptimality conditions.
Keywords
Sufficient optimality condition Optimal shape control Dual dynamic programming1 Introduction
We consider an optimum design problem with the aim to determine the best location of hollows γ in a given bounded subdomain ω surrounded by the exterior subdomain Ω in \(\mathbb {R}^{2}\) with smooth boundaries ∂ω, Γ = ∂Ω∖∂ω. In the interior subdomain ω the physical phenomenon are described by the linear PDE and in the exterior domain the processes are governed by nonlinear PDE subject to some external function. We took that problem from the paper (Szulc and Zochowski 2015) where more details is presented. The design problem can be considered as a two level optimization problem. At the first level an optimal control within a set of admissible controls is determined for a given location of the source. At the second level an optimal location of the source in terms of its characteristic function is selected in such a way that the resulting value of the cost functional is the best possible within the set of admissible locations. The problem considered in this paper is for an elliptic equation. The problem at the second level is nonconvex, which leads to well known difficulties with the solution procedure. In particular, such difficulties and possible relaxation procedures are discussed e.g. in Kohn and Strang (1986) and Murat and Tartar (1985) mainly from the point of view of existence of solutions in optimum design problems. The standard techniques in classical optimal control theory are based on the lower semicontinuity of some physical quantity (functional) with respect to control and on the compactness of the set of admissible controls. In the optimum design problem the location of the source is optimized. Thus, the lower semicontinuity of the shape functional is required with respect to some family of sets. So, except for very particular cases, there is no optimal location in optimum design problem (see Delfour and Zolesio 2001). That is why many authors confine themself only to numerical analyzis and this is just done in Sokolowski and Zochowski (1999) applying topological derivative and in Szulc and Zochowski (2015). In the last decade some optimum design problems were investigated for timedependent state equations, including the wave equation or the heat equation (see e.g. Hebrard and Henrot 2003; Münch 2008; Periago 2009; Nowakowski and Sokolowski 2012). Our aim is also to apply the particular variational method, widely used in the classical optimal control theory, which is the dynamic programming technique. In this way, we need neither relaxation nor homogenization of the problem under investigation. We also develop a new computational technic basing on dual dynamic method to characterize the best approximate location of the source explicitly, which is useful for possible applications. In the paper we assume that the number of hollow voids γ is finite but not fixed and non zero i.e. γ = γ_{1} ∪ ... ∪ γ_{n}, n ∈N, γopen \( \varsubsetneq \omega \) and such that 𝜖_{2} ≥ vol γ ≥ 𝜖_{1} > 0, with given 𝜖_{1}, 𝜖_{2}. Moreover we assume that all boundaries of γ are smooth. Put D = Ω ∪ ω.
Our main aim is to be able to characterize the optimal (ε optimal) pair \((\bar {u}\), \(\bar {\gamma })\) and the εminimal value of J. Note, that \(\bar {\gamma }\) is an optimal (or ε optimal) open set in our notation. We call a pair (u(⋅), γ) ”admissible” if it satisfies (2)–(5) and the conditions imposed on γ. Then the corresponding trajectory u(⋅) is said to be admissible and γ is an admissible set. The set of all admissible pairs is denoted by Ad.
The problems with unknown a subset γ appear in many papers. In Münch (2008) the optimum design for two dimensional wave equation is studied, in Münch (2009) an optimal location of the support of the control for onedimensional wave equation is determined, in Hebrard and Henrot (2005) and Hebrard and Henrot (2003) the optimal geometry for controls in stabilization problem is considered. In all mentioned papers different approaches to the design problems have been investigated, and some numerical results are presented. The existence of optimum design is essential if we have not at hand any sufficient optimality conditions. From the beginning of the last century, under strong influence of Hilbert, the existence issue became one of the fundamental questions in many branches of mathematics, especially in calculus of variations as well as in its branch, the optimal control theory. Of course, following the existence proof, the next step is derivation of necessary optimality conditions and the evaluation of the minimum argument. However, it should be pointed out that for many variational problems the existence of a solution accompanied by some necessary optimality conditions are not sufficient to find the argument of minimum in practice. On the other hand, having at hand a stronger result, i.e. the sufficient optimality conditions for a minimum in a specific problem, replaces the requirement for the existence.
In the present paper the framework of dynamic programming together with sufficient optimality conditions (the socalled verification theorem for relative minimum) is proposed for a solution of the optimum design problem. Different approaches are given in Sokolowski and Zochowski (1999), Bednarczuk et al. (2000), Nowakowski and Sokolowski (2012), Fulmański et al. (2014), and Fulmański and Nowakowski (2014). In Sokolowski and Zochowski (1999), Bednarczuk et al. (2000), and Nowakowski and Sokolowski (2012) the shape problem is formulated in terms of characteristic functions which define the support of control while in Fulmański et al. (2014) and Fulmański and Nowakowski (2014) also the framework of dynamic programming together with sufficient optimality conditions is proposed for a solution of the optimum design problem. In Fulmański et al. (2014) the problem of dividing tube is investigated by reduction of the shape optimization problem to a classical control problem. To do that the authors apply the level set approach to build deformation, following Zolésio (Sokolowski and Zolesio 1992) they introduce a field depending on control, which define the type of deformation, and formulate the control problem governed by ordinary differential equation (defined by field). Next the classical dynamic programming is developed for such a problem. In Fulmański and Nowakowski (2014) a little similar approach by level set method is applied to the optimization problem of deformable structures described by NavierStock equation but then the dual dynamic programming is delveloped to derive the verification theorem (again for dynamics governed by ordinary differential equation). In both papers completely different numerical algorithms are constructed the base of which are different approximate verification theorems. However we should stress that approaches based on the level set method described in Fulmański et al. (2014) and Fulmański and Nowakowski (2014) use smaller set of admissible deformation (described by a field depending on control) i.e. in our case a significantly smaller set of the sources γ. In the presentation (Kaźmierczak 2008) similar ideas to those of Fulmański et al. (2014) and Fulmański and Nowakowski (2014) are mentioned but nothing is precisely formulated and proved. Our goal is not the standard analysis of the problem as e.g. in Delfour and Zolesio (2001) or Fulmański et al. (2014) and Fulmański and Nowakowski (2014) but the approximate solution by application of the sufficient ε optimality conditions given by the dual dynamic programming directly to our problem (P). That means we admit full set of admissible sources γ. This approach seems to be new and the result obtained is original, to our best knowledge.
We provide a dual dynamic programming approach to control problems (2)–(5). This approach allows us to obtain the sufficient conditions for optimality in problem considered. We believe that the conditions for problems of type (P) in terms of the dual dynamic programming, that we formulate here, have not been provided earlier. There are two main difficulties that must be overcome in such problems as (P). The first one consists in the following observation. We have no possibility to perform perturbations of the problem  as it is considered in the fixed set D with boundary condition (5)  which can be compared to the onedimensional case given in Bellman (1957) and Fleming and Rishel (1975). The second one is that we deal with elliptic equation for state and controls (2 ). The technique we apply is similar to the methods from Galewska and Nowakowski (2006) and Nowakowski (1992). The main idea of the methods from Galewska and Nowakowski (2006) and Nowakowski (1992) is that they carry over all objects used in dynamic programming to dual space  space of multipliers (similar to those which appear in the Pontryagin maximum principle). Next, instead of classical value function (which for problem (P) makes no sense), we define an auxiliary function V (x, p) satisfying the second order partial differential equation of dual dynamic programming (compare Galewska and Nowakowski 2006). Investigations of the properties of this function lead to an appropriate verification theorem. We introduce also the concept of an optimal dual feedback control and provide new sufficient ε optimality conditions determined within our framework. Just by using the approximate differential equation of dual dynamic programming (9 ) (see below) and εoptimal dual feedback control (section 4) we are able to solve problem (P) completely, from the approximate point of view i.e we find aproximate location of γ_{ε} as well for εoptimal value of J.
2 Dual dynamic programming approach
2.1 A verification theorem
Theorem 1
Proof
2.2 Optimal dual feedback control
This section is devoted to the notion of an optimal dual feedback control. We present appropriate definitions and the sufficient conditions for optimality which follow from the verification theorem.

for \(\left \vert \frac {y}{y^{0}}\right \vert \geq 1\)a function χ(⋅,⋅)which to those (x, p) ∈ Passigns an admissible set γ_{p} ⊂ ω, x ∈ γ_{p}, such that χ(x, p) = 0,

for \(\left \vert \frac {y}{y^{0}}\right \vert <1\)a function χ(⋅,⋅)which to those (x, p) ∈ Passigns an admissible set γ_{p} ⊂ ω , x∉γ_{p}, such that χ(x, p) = 1 .
Thus, for a given p only one γ_{p} is assigned. This means that χ(x, p) = 0 for \(\left \vert \frac {y}{y^{0}}\right \vert \geq 1\), χ(x, p) = 1 for \(\left \vert \frac {y}{y^{0}}\right \vert <1\).
Now, we formulate and prove the sufficient optimality conditions for the existence of an optimal dual feedback control, again in terms of the auxiliary function V (x, p).
Theorem 2
Proof
3 Approximate optimality for the problem (P)
3.1 εoptimality
Now we are ready to define notions of εoptimal dual feedback controls χ_{ε}(x, p) of εoptimal pair \((\bar {u}_{\varepsilon }(\cdot ),\bar {\gamma }_{\varepsilon })\).
Theorem 3
Proof
3.2 Numerical algorithm
The verification theorem formulated in the former section allows us to build numerical approach to calculate suboptimal pair \((\bar {u}_{\varepsilon }(\cdot ),\bar {\gamma }_{\varepsilon })\) such that \(S_{\varepsilon D}^{\bar {\mathbf {u}}_{\varepsilon }\bar {y}_{\varepsilon }^{0}}\) satisfies (33). The algorithm we present below ensure that we find in finite number of steps suboptimal pair.
 1.
 2.Form \(Ad_{\bar {u}_{\varepsilon }}\) as a finite family of N pairs (u(⋅), γ) :
 (a)
Define sets γ_{n} in ω, n = 1,..., N.
 (b)
To calculate u_{n}, n = 1,..., N solve inequality (28).
 (a)
 3.
Find minimal value of J(γ_{n}), n = 1,..., N and corresponding to it pair denote by \((\hat {u}(\cdot ),\hat {\gamma })\).
 4.Assume any fixed \(\bar {y}_{\varepsilon }^{0}<0\) and determine \(\hat {y}(\cdot )\) from the relation$$\hat{u}(x)=\tilde{V}_{y}(x,\bar{y}_{\varepsilon }^{0},\hat{y}(x)). $$
 5.
3.3 Scalability and convergence of the algorithm
First we would like to stress that using any package to solve nonlinear differential equation we are able only to find approximate solution i.e. the numerical solutions do not satisfy in our case the Eqs. 2, 9. Therefore any sufficient optimality conditions for approximate optimality should take into account those circumstances. In our algorithm we do it, i.e. we solve with any computational package the inequalities (28), (26) and check whether the calculated solutions satisfy (28), (26) with given ε. The number N in points 2., 3. of the above Algorithm may be arbitrarily large. We can then divide N on smaller parts and do calculations for (28) and point 3. (on those parts) independly (on different processors) and next to find minimal value of J as a minimum of minimal values of J on those parts. If we were looking for γ being balls (or systems of balls) then we can cover the whole bounded domain ω with finite number of balls of any positive fixed radius. Then the convergence of the algorithm is obvious. However if we are interested not only in locations but also in shapes of γ then from the theoretical point of view the convergence of algorithm is still almost obvious as the closure of ω is a compact set, thus there is always finite covering of the closure of ω by open sets γ. Hence we can calculate at least theoreticaly the best of them using te above algorithm and any other set γ, by continuity of our functional is near one of a set from the covering family.
3.4 The parameters
 one ballwhere \((\bar x_{1}, \bar x_{2})\)  the center of γ.$$\gamma = \{(x_{1},x_{2}) \in \omega: (x_{1}  \bar x_{1})^{2} + (x_{2}  \bar x_{2})^{2} < (\bar{\varepsilon })^{2} \}, $$
 two ballswhere \((\bar {x_{1}^{1}}, \bar {x_{2}^{1}})\)  the center of γ_{1},$$\gamma_{1} = \{(x_{1},x_{2}) \in \omega: (x_{1}  \bar {x_{1}^{1}})^{2} + (x_{2}  \bar {x_{2}^{1}})^{2} < (\bar{\varepsilon })^{2} \}, $$where \((\bar {x_{1}^{2}}, \bar {x_{2}^{2}})\)  the center of γ_{2}, γ = γ_{1} ∪ γ_{2} and γ_{1} ∩ γ_{2} = ∅.$$\gamma_{2} = \{(x_{1},x_{2}) \in \omega: (x_{1}  \bar {x_{1}^{2}})^{2} + (x_{2}  \bar {x_{2}^{2}})^{2} < (\bar{\varepsilon })^{2} \}, $$
 three ballswhere \((\bar {x_{1}^{1}}, \bar {x_{2}^{1}})\)  the center of γ_{1},$$\gamma_{1} = \{(x_{1},x_{2}) \in \omega: (x_{1}  \bar {x_{1}^{1}})^{2} + (x_{2}  \bar {x_{2}^{1}})^{2} < (\bar{\varepsilon })^{2} \}, $$where \((\bar {x_{1}^{2}}, \bar {x_{2}^{2}})\)  the center of γ_{2},$$\gamma_{2} = \{(x_{1},x_{2}) \in \omega: (x_{1}  \bar {x_{1}^{2}})^{2} + (x_{2}  \bar {x_{2}^{2}})^{2} < (\bar{\varepsilon })^{2} \}, $$where \((\bar {x_{1}^{3}}, \bar {x_{2}^{3}})\)  the center of γ_{3}, γ = γ_{1} ∪ γ_{2} ∪ γ_{3}, γ_{1} ∩ γ_{2} = ∅, γ_{1} ∩ γ_{3} = ∅ and γ_{2} ∩ γ_{3} = ∅.$$\gamma_{3} = \{(x_{1},x_{2}) \in \omega: (x_{1}  \bar {x_{1}^{3}})^{2} + (x_{2}  \bar {x_{2}^{3}})^{2} < (\bar{\varepsilon })^{2} \}, $$
We assume that ω_{ε} = ω ∖ γ, ε = 0.1, \(\hat {\gamma }\)  the optimal γ, \(\bar {y}_{\varepsilon }^{0} = 0.005\) and N = 500.

function z_{d} on D ∖ γ_{zd}, where \(\gamma _{zd} = \{(x_{1},x_{2}) \in \omega : (x_{1} + 0.2)^{2} + (x_{2}  0.2)^{2} < (\bar {\varepsilon })^{2} \}\) for the first case,

functions z_{d1} and z_{d2} on D ∖ γ_{zd12}, where \(\gamma _{zd12} = \{(x_{1},x_{2}) \in \omega : (x_{1}  0.2)^{2} + (x_{2} + 0.1)^{2} < (\bar {\varepsilon })^{2} \} \cup \{(x_{1},x_{2}) \in \omega : (x_{1} + 0.1)^{2} + (x_{2}  0.2)^{2} < (\bar {\varepsilon })^{2} \}\) for the second case,

functions z_{d1} and z_{d2} on D ∖ γ_{zd12}, where \(\gamma _{zd12} = \{(x_{1},x_{2}) \in \omega : (x_{1}  0.2)^{2} + (x_{2} + 0.1)^{2} < (\bar {\varepsilon })^{2} \} \cup \{(x_{1},x_{2}) \in \omega : (x_{1} + 0.1)^{2} + (x_{2}  0.2)^{2} < (\bar {\varepsilon })^{2} \} \cup \{(x_{1},x_{2}) \in \omega : (x_{1} + 0.2)^{2} + (x_{2} + 0.2)^{2} < (\bar {\varepsilon })^{2} \}\) for the third case.
We are looking for optimal control as ball (or systems of balls) that generate minimal values of J and satisfy (34)–(35).
3.5 Numerical calculations
 1.
Let ε = 0.1.
 2.
Let fixed number of balls (one, two or three).
 3.We calculate function z_{d} (or z_{d1} and z_{d2} respectively). z_{d} is a solution of the following boundary value problem:$${\Delta} z_{d}(x) + \chi ({\Omega} ){z_{d}^{3}}(x)=\chi ({\Omega} ) f_{zd}(x),\ \ \text{ \ }x\in D\backslash \gamma_{zd}, $$$$z_{d}(x)= 0,\text{ \ }x\in {\Gamma} , $$where f_{zd}(x) = x_{1} for x ∈ D∖γ_{zd} when we consider one ball.$$\partial_{n} z_{d}(x)= 0\text{ \ on \ }\partial \gamma_{zd}, $$
 4.
If we consider two or three balls we solve above boundary value problem due to two functions f_{zd1}(x) = x_{1} + x_{2} for x ∈ D∖γ_{zd12} and f_{zd2}(x) = x_{1} − x_{2} for x ∈ D∖γ_{zd12}.
 5.
For given parameters we randomly generate γ for fixed \(\bar {\varepsilon }\).
 6.For given γ, ω and Ω we solve the following systems of equations$${\Delta} u(x)=F(x,u(x)),\ \ \text{ \ }x\in D\backslash \gamma, $$$$u(x)= 0,\text{ \ }x\in {\Gamma} , $$$$u(x)=\varphi (x),\text{ }\partial_{n}u(x)=\partial_{n}\varphi (x),\text{ } x\in \partial \omega, $$where \(\omega \varsubsetneq {\Omega }\).$$\partial_{n}u(x)= 0\text{ \ on \ }\partial \gamma, $$where \(f:{\Omega } \mapsto \mathbb {R}\) defined as$$F(x,u(x))=\left\{ \begin{array}{c} u^{3}(x)+f(x),\text{ \ }x\in {\Omega}, \\ \text{ \ }0,\text{ \ \ \ \ \ \ \ \ \ \ \ \ \ \ }x\in \omega \backslash \gamma, \end{array} \right. $$$$f(x) = x_{1}+x_{2}, \text{ \ }x = (x_{1},x_{2})\in {\Omega}. $$
 7.We calculate the value of (1)for one ball and$$ J(\gamma )=\frac{1}{2}\int\nolimits_{\Omega}(u(x)z_{d}(x))^{2}dx. $$(38)for two and three balls.$$ J(\gamma )=\frac{1}{2}\int\nolimits_{\Omega}(u(x)z_{d1}(x))^{2}dx+\frac{1}{2}\int\nolimits_{\Omega}(u(x)z_{d2}(x))^{2}dx. $$(39)
 8.
We repeat 5.  7. N = 500times.
 9.
We select the minimum value among all values from (1). It means that we find minimal value of J(γ_{n}), n = 1,..., N and corresponding to it pair denote by \((\hat {u}(\cdot ),\hat {\gamma })\). minJ for the optimal \((\hat {u}(\cdot ),\hat {\gamma })\) defined by the \(\hat {\gamma }\).
 10.We generate the set$$\boldsymbol{Y}=\{(y^{0},y)=p; \text{ there exists }\ (x,p)\in P \}:$$$$\boldsymbol{Y} = \{(0.005, 0.110100), $$$$(0.005, 0.060100 ), (0.005, 0.010100), $$$$(0.005, 0.039900), (0.005, 0.089900),$$$$(0.005, 0.001000), (0.005, 0.001000) \}. $$
 11.
 12.We assume \(y_{\varepsilon }^{0} = 0.005\) and determine \(\hat {y}(\cdot )\) from the relation$$\hat{u}\left( t ,x\right) = \hat{V}_{y}(t ,x , 0.005 ,\hat{y}(t ,x)). $$
 13.
For \(\tilde {V}\), \((\hat {u} (\cdot ) ,\hat {\gamma } (\cdot ))\) we check the inequality (34), and we obtain that \(\tilde {V}\) and \((\hat {u}(\cdot ),\hat {\gamma })\) satisfy (34)–(35) so \((\hat {u}(\cdot ),\hat {\gamma })\) is an εoptimal pair and\(\ J(\hat {\gamma })\) is an ε optimal value with ε = 0.1.
3.6 Numerical examples
We present four examples, each for an otherwise defined control.
3.6.1 Example 1

We fixed ε = 0.1.

From 2. we calculate function z_{d} (Fig. 2).
 For given parameters we repeat 3.  5. N = 500times. We select the minimum value among all values from (1). It means that we find minimal value of J(γ_{n}), n = 1,..., N and corresponding to it pair denote by \((\hat {u}(\cdot ),\hat {\gamma })\). (Figs. 3, 4 and 5)for the optimal \((\hat {u}(\cdot ),\hat {\gamma })\) defined by the center of \(\hat {\gamma }\), where$$\min J = 0.022086 $$$$(\hat x_{1}, \hat x_{2}) = (0.2012,0.1934). $$

We generate the set Y
 We assume \(y_{\varepsilon }^{0} = 0.005\) and determine \(\hat {y}(\cdot )\) from the relation$$\hat{u}\left( t ,x\right) = \hat{V}_{y}(t ,x , 0.005 ,\hat{y}(t ,x)). $$

For \(\tilde {V}\), \((\hat {u} (\cdot ) ,\hat {\gamma } (\cdot ))\) we check the inequality (34), and we obtain that \(\tilde {V}\) and \((\hat {u}(\cdot ),\hat {\gamma })\) satisfy (34)–(35) so \((\hat {u}(\cdot ),\hat {\gamma })\) is an εoptimal pair and\(\ J(\hat {\gamma })\) is an ε optimal value with ε = 0.1.
3.6.2 Example 2
 for the optimal \((\hat {u}(\cdot ),\hat {\gamma })\) (Fig. 8) defined by the center of \(\hat {\gamma }\), where$$\min J = 0.041088 $$$$(\bar {x_{1}^{1}}, \bar {x_{2}^{1}}) = (0.2100, 0.1028), $$$$(\bar {x_{1}^{2}}, \bar {x_{2}^{2}}) = (0.0982, 0.1863). $$
3.6.3 Example 3
 for the optimal \((\hat {u}(\cdot ),\hat {\gamma })\) defined by the center of \(\hat {\gamma }\), where$$\min J = 0.022472 $$$$(\bar {x_{1}^{1}}, \bar {x_{2}^{1}}) = (0.2156, 0.1012), $$$$(\bar {x_{1}^{2}}, \bar {x_{2}^{2}}) = (0.0862, 0.2243), $$$$(\bar {x_{1}^{3}}, \bar {x_{2}^{3}}) = (0.1868, 0.2011). $$
3.6.4 Example 4
Let us γ as one hole defined as nonconvex set  two balls with nonempty intersection.
Remark 1
In step 2. we took, for simplicity, balls for ω as in final calculations the obtained results turn out satisfying and computations were significantly shorter.
Remark 2
In Example 4 we considered the controls as nonconvex holes. To calculate function z_{d} we defined domain with one hole defined as a ball. For this case the verification theorem is satisfied.
4 Conclusions
The dual dynamic programming by construction furnishes the sufficient optimality conditions, therefore it is a powerful tool for solution of optimization problems which enjoy the special structure. This is especially important when we deal with approximate (numerical) solutions. In the paper for the optimum design problem the dual dynamic programming and dual feedback notions are developed as well their approximate counterparts. Next the solutions of that problem are characterized in terms of verification theorems. The approximate verification theorem allows us to calculate approximate solution but also to state how far we are from the inf J(γ). In Szulc and Zochowski (2015) the value of calculated numerically functional is almost the same (we took the same parameters in our example) but the authors in Szulc and Zochowski (2015) can only assert that it is some step of calculation of the minimal value. In a subsequent paper we are going to apply the dynamic programming technique to more general shape optimization problems.
Notes
References
 Bednarczuk E, Pierre M, Rouy E, Sokolowski J (2000) Tangent sets in some functional spaces. Nonlinear Anal. 42, no. 5, Ser A: Theory Methods, 871–886Google Scholar
 Bellman R (1957) Dynamic programming. Princeton University Press, PrincetonzbMATHGoogle Scholar
 Delfour MC, Zolesio JP (2001) Shapes and geometries  analysis, differential calculus and optimization, advances in design and control. SiamGoogle Scholar
 Fleming WH, Rishel RW (1975) Deterministic and stochastic optimal control. Springer, New YorkCrossRefzbMATHGoogle Scholar
 Fulmański P, Nowakowski A (2014) Dynamic programming approach to shape optimization. Control Cybernet 43(3):379–401MathSciNetzbMATHGoogle Scholar
 Fulmański P, Nowakowski A, Pustelnik J (2014) Dynamic programming approach to structural optimization problem—numerical algorithm. Opuscula Math 34(4):699–724MathSciNetCrossRefzbMATHGoogle Scholar
 Galewska E, Nowakowski A (2006) A dual dynamic programming for multidimensional elliptic optimal control problems. Numer Funct Anal Optim 27(34):279–289MathSciNetCrossRefzbMATHGoogle Scholar
 Hebrard P, Henrot A (2005) Spillover phenomenon in the optimal locations of actuators. SIAM J Control Optim 44:349–366MathSciNetCrossRefzbMATHGoogle Scholar
 Hebrard P, Henrot A (2003) Optimal shape and position of the actuators for the stabilization of a string. Syst Control Lett 48:199–209MathSciNetCrossRefzbMATHGoogle Scholar
 Kaźmierczak A (2008) Dual dynamic programming approach to shape optimization for NavierStokes equations, Presentation, www.ibspan.waw.pl/OPTPDE2008/prezentacje/AKaz.pdf
 Kohn RV, Strang G (1986) Optimal design and relaxation of variational problems. Comm Pure Appl Math 39:113–182MathSciNetCrossRefzbMATHGoogle Scholar
 Murat F, Tartar L (1985) Calcul des Variations et Homog ´ en ´ eisation, Les M ´ ethodes d’Homogeneisation Theories et Applications en Physique, Coll. Dir. Etudes et Recherches EDF, 57, Eyrolles, Paris, 319–369Google Scholar
 Münch A (2009) Optimal location of the support of the control for the 1D wave equation: numerical investigations. Comput Optim Appl 42(3):443–470MathSciNetCrossRefzbMATHGoogle Scholar
 Münch A (2008) Optimal design of the support of the control for the 2D wave equation: a numerical method. Int J Numer Anal Model 5(2):331–351MathSciNetzbMATHGoogle Scholar
 Nowakowski A (1992) The dual dynamic programming. Proc Am Math Soc 116(4):1089–1096MathSciNetCrossRefzbMATHGoogle Scholar
 Nowakowski A, Sokolowski J (2012) On dual dynamic programming in shape control. Commun Pure Appl Anal 11:2473–2485MathSciNetCrossRefzbMATHGoogle Scholar
 Periago F (2009) Optimal shape and position of the support for the internal exact control of a string. Syst Control Lett 58:136–140MathSciNetCrossRefzbMATHGoogle Scholar
 Sokolowski J, Zolesio JP (1992) Introduction to shape optimiation. Springer, BerlinCrossRefzbMATHGoogle Scholar
 Sokolowski J, Zochowski A (1999) Topological derivative for elliptic problems. Inverse Prob 15:123–134MathSciNetCrossRefzbMATHGoogle Scholar
 Szulc K, Zochowski A (2015) Application of topological derivative to accelerate genetic algorithm in shape optimization of coupled models. Struct Multidisc Optim 51:183–192MathSciNetCrossRefGoogle Scholar
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