E. Casas, F. Troltzsch and A. Unger, \Second order sufficient optimality conditions for some state constrained control problems of semilinear elliptic equations", 21 Fakultat fur Mathematik, Technische Universitat Chemnitz, Preprint 97-19, to appear in SIAM J. Control Optim.  Home/Search   Document Details and Download   Summary   Related Articles   Check

....function increases at the same rate that the square of distance to the solution) For a space dimension larger than 3, the sufficient conditions are formulated in terms of two norms for the control space, i.e. L 2( Omega Gamma and L s( Omega Gamma for some s n=2. This may be compared to e.g. [9], where more general nonlinearities are considered, but only sufficient conditions for quadratic growth, using the L 1( Omega Gamma3 are given. In section 3, assuming a weak second order sufficient condition to hold, and the feasible set to be polyhedric, we provide a formula for computing the ....

....unconstrained problems. The extension to polyhedric constraint sets is discussed in [16, 17, 18] assuming stronger second order conditions, that extend the strong regularity condition of [21] Related results, in the context of optimal control of partial differential equation, may be found in [9, 10]. These papers consider more general nonlinearities and, therefore, use the two norms approach with spaces L 2 ( Omega Gamma and L 1( Omega Gamma9 Note also that there is no characterization of quadratic growth in these papers. INRIA Second order analysis for control constrained optimal ....

E. Casas, F. Tr oltzsch and A. Unger, Second order sufficient optimality conditions for a nonlinear elliptic boundary control problem, Zeitschrift fur Analysis und ihre Anwendungen 15 (1996), pp. 687--707.

.... in the sense of L 1 ( Sigma) this can be even proved in L p ( Sigma) for p N 1) For a weaker version of second order sufficient conditions we refer to the proof in Goldberg and Troltzsch [13] In the case of an elliptic equation of state this is shown in Casas, Troltzsch and Unger [8]. Their technique can easily be transferred to the parabolic case considered here. 3 The SQP Method In this section we recall the (continuous) SQP method. Let w 0 = y 0 ; p 0 ; u 0 ) be a starting triplet (we shall assume that w 0 is close to the reference triplet w = y; u; p) Then the ....

Casas, E., Troltzsch, F. and Unger,A (1996), "Second order sufficient optimality conditions for a nonlinear elliptic control problem," J. of Analysis and Applications (ZAA) 15, 687--707.

....simplifies in an obvious way. One can show that ( y; u) is locally optimal for the control problem in the sense of Y Theta L 1 ( Sigma) if ( y; u; p) satisfies the first order optimality system together with (SSC) We refer to the elliptic case discussed in Casas, Troltzsch and Unger [7]. The extension to the parabolic case is straightforward. 4 The Generalized Newton method Completely analogous to (2.4) we can formulate the adjoint equation (3.1) as p = G T (y(T ) Gamma y T ) DB 0 (y)p; 4.1) where the final operator G T : C( Omega Gamma Y gives the solution of (3.1) ....

E. Casas, A. Unger, and F. Troltzsch. Second order sufficient optimality conditions for a nonlinear elliptic control problem. J. Analysis and its Appl. (ZAA), 15, No.3:687-- 707, 1996.

....to the theory of second order sufficient optimality conditions for optimal control problems governed by nonlinear partial differential equations. We consider the control of semilinear parabolic equations with pointwise constraints on the control and the state. Recently, Casas, Troltzsch and Unger [6] have discussed second order sufficient conditions for the boundary control of semilinear elliptic equations with pointwise state constraints. It is convenient, to formulate this class of constraints in spaces of continuous functions, hence the associated Lagrange multipliers are Borel measures. ....

....although the basic difficulty of low regularity cannot be entirely solved. The theory for parabolic equations differs from the elliptic case mainly in the regularity of the solutions, while many other aspects are identical. In view of this, we shall heavily rely on the results presented in [6]. Some proofs can be adopted word for word from associated theorems stated therein. Hence we will concentrate on specific features of parabolic problems rather than to repeat lengthy constructions being analogous to [6] This research was partially supported by the European Union, under the HCM ....

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Casas, E., Troltzsch, F., Unger, A.: Second order sufficient optimality conditions for some state-constrained control problems of semilinear elliptic equations, to appear in SIAM J. Control Optim.

....the solutions of partial differential equations with measures as data. Some important questions are still unsolved. We refer, for instance, to the discussion of second order sufficient optimality condition for elliptic problems with state constraints investigated by Casas, Troltzsch and Unger [7]. 2. The optimal control problem. Our control system is given by the semilinear parabolic initial boundary value problem 8 : y t div(A grad y) d y y = e Q d v v in Q y b y y = e Sigma b u u on Sigma y(0) e Omega dw w in Omega : 2.1) We consider this equation of ....

E. CASAS, F. TR OLTZSCH, and A. UNGER, Second order sufficient optimality conditions for some state-constrained control problems of semilinear elliptic equations, to appear in SIAM J. Control Optim.

....elliptic differential equations. We want to prove only the existence of a solution. For first order necessary optimality conditions we refer for instance to Casas (1993) and Bonnans and Casas (1991) Moreover, a new development in second order optimality conditions occurs (consult, for instance, Casas, Troltzsch and Unger, 1996). The questions concerning optimality conditions requiring assumptions on differentiability will not be considered here. As mentioned above, a standard method for the proof of existence is known. This method was applied, for instance, in Lions (1968) Zeidler (1990) and in Casas (1993) The main ....

Casas, E., Tr oltzsch, F. and Unger, A. (1996) Second order sufficient optimality conditions for a nonlinear elliptic boundary control problem.

....state constraints of equality and inequality type are given. First order necessary optimality conditions are already well known for this type of problems (cf. Bonnans and Casas [3] and we derive them only for convenience. In contrast to this, it seems to the authors that up to now only the paper [8] deals with second order conditions for elliptic control problems. In that work, sufficient optimality conditions were derived for the case without state constraints, which are in some sense arbitrarily close to the corresponding necessary ones. We aim to extend these conditions to problems with ....

E. Casas, F. Tr oltzsch, and A. Unger, Second order sufficient optimality conditions for a nonlinear elliptic control problem, J. for Analysis and its Applications (ZAA) 15 (1996), pp. 687--707.

No context found.

E. Casas, F. Troltzsch and A. Unger, \Second order sufficient optimality conditions for some state constrained control problems of semilinear elliptic equations", 21 Fakultat fur Mathematik, Technische Universitat Chemnitz, Preprint 97-19, to appear in SIAM J. Control Optim.

No context found.

E. Casas, F. Troltzsch and A. Unger, \Second order sufficient optimality conditions for a nonlinear elliptic control problem", J. for Analysis and its Applications, vol 15, pp. 687-707, 1996.

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