Determination of optimal feedback terminal controllers for general boundary conditions using generating functions

Chandeok Park, Daniel J. Scheeres

Research output: Contribution to journalArticlepeer-review

71 Citations (Scopus)


Given a nonlinear system and a performance index to be minimized, we present a general approach to expressing the finite time optimal feedback control law applicable to different types of boundary conditions. Starting from the necessary conditions for optimality represented by a Hamiltonian system, we solve the Hamilton-Jacobi equation for a generating function for a specific canonical transformation. This enables us to obtain the optimal feedback control for fundamentally different sets of boundary conditions only using a series of algebraic manipulations and partial differentiations. Furthermore, the proposed approach reveals an insight that the optimal cost functions for a given dynamical system can be decomposed into a single generating function that is only a function of the dynamics plus a term representing the boundary conditions. This result is formalized as a theorem. The whole procedure provides an advantage over methods rooted in dynamic programming, which require one to solve the Hamilton-Jacobi-Bellman equation repetitively for each type of boundary condition. The cost of this favorable versatility is doubling the dimension of the partial differential equation to be solved.

Original languageEnglish
Pages (from-to)869-875
Number of pages7
Issue number5
Publication statusPublished - 2006 May

Bibliographical note

Funding Information:
The authors acknowledge support from NSF Grant CMS-0408542.

All Science Journal Classification (ASJC) codes

  • Control and Systems Engineering
  • Electrical and Electronic Engineering


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