Solution differentiability for parametric nonlinear control problems with control-state constraints View Full Text


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Article Info

DATE

1995-08

AUTHORS

H. Maurer, H. J. Pesch

ABSTRACT

This paper considers parametric nonlinear control problems subject to mixed control-state constraints. The data perturbations are modeled by a parameterp of a Banach space. Using recent second-order sufficient conditions (SSC), it is shown that the optimal solution and the adjoint multipliers are differentiable functions of the parameter. The proof blends numerical shooting techniques for solving the associated boundary-value problem with theoretical methods for obtaining SSC. In a first step, a differentiable family of extremals for the underlying parameteric boundary-value problem is constructed by assuming the regularity of the shooting matrix. Optimality of this family of extremals can be established in a second step when SSC are imposed. This is achieved by building a bridge between the variational system corresponding to the boundary-value problem, solutions of the associated Riccati ODE, and SSC.Solution differentiability provides a theoretical basis for performing a numerical sensitivity analysis of first order. Two numerical examples are worked out in detail that aim at reducing the considerable deficit of numerical examples in this area of research. More... »

PAGES

285-309

References to SciGraph publications

  • 1990-01. Sensitivity analysis of optimization problems in Hilbert space with application to optimal control in APPLIED MATHEMATICS & OPTIMIZATION
  • 1970-04. Sequential gradient-restoration algorithm for optimal control problems in JOURNAL OF OPTIMIZATION THEORY AND APPLICATIONS
  • 1990-12. New general guidance method in constrained optimal control, part 2: Application to space shuttle guidance in JOURNAL OF OPTIMIZATION THEORY AND APPLICATIONS
  • 1990-12. New general guidance method in constrained optimal control, part 1: Numerical method in JOURNAL OF OPTIMIZATION THEORY AND APPLICATIONS
  • 1995-05. Optimality, stability, and convergence in nonlinear control in APPLIED MATHEMATICS & OPTIMIZATION
  • 1970-11. Modified quasilinearization and optimal initial choice of the multipliers part 2—Optimal control problems in JOURNAL OF OPTIMIZATION THEORY AND APPLICATIONS
  • 1989-08. Sufficient optimality conditions for nonconvex control problems with state constraints in JOURNAL OF OPTIMIZATION THEORY AND APPLICATIONS
  • 1981. First and second order sufficient optimality conditions in mathematical programming and optimal control in MATHEMATICAL PROGRAMMING AT OBERWOLFACH
  • 1990. Optimal control problems under disturbances in SYSTEM MODELLING AND OPTIMIZATION
  • 1995-09. Second-order sufficient conditions for control problems with mixed control-state constraints in JOURNAL OF OPTIMIZATION THEORY AND APPLICATIONS
  • 1988-08. Another Jacobi sufficiency criterion for optimal control with smooth constraints in JOURNAL OF OPTIMIZATION THEORY AND APPLICATIONS
  • 1983. Perturbations, Approximations and Sensitivity Analysis of Optimal Control Systems in NONE
  • 1992-01. Second-order conditions and constraint qualifications in stability and sensitivity analysis of solutions to optimization problems in Hilbert spaces in APPLIED MATHEMATICS & OPTIMIZATION
  • 1989-07. Sufficiency conditions with minimal regularity assumptions in APPLIED MATHEMATICS & OPTIMIZATION
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    http://scigraph.springernature.com/pub.10.1007/bf02192081

    DOI

    http://dx.doi.org/10.1007/bf02192081

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