An Optimization Problem in Control Theory

  • Ker-I Ko
Part of the Progress in Theoretical Computer Science book series (PTCS)


The following optimization problem has arisen from the non-classical control theory: for a given continuous function f : [0, 1]4 → [0, 1] that satisfies the Lipschitz condition, compute the minimum value
$${{\alpha }^{*}}(f) = \mathop{{\inf }}\limits_{{{{g}_{1}},{{g}_{2}} \in C[0,1]}} \int_{0}^{1} {\int_{0}^{1} {f({{x}_{1}},{{x}_{2}},{{g}_{1}}({{x}_{1}}),g({{x}_{2}}))d{{x}_{1}}d{{x}_{2}}.} }$$
Intuitively, the function f may be viewed as the cost function on inputs x1, x2 ∈ [0, 1] and the corresponding decisions g1(x) and g2(x2) on these inputs. The decision functions g1 and g2 are based only on part of the input values and perform, in a sense, in the distributed manner. The goal is to find the best distributed decision functions that minimize the average cost f. The condition that f satisfies the Lipschitz condition is necessary so that the minimum value α*(f) is computable. For more about the motivations and applications of this problem, see Papadimitriou and Tsitsiklis [1982, 1986].


Lipschitz Condition Boolean Variable Exponential Time Truth Assignment Follow Optimization Problem 
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Copyright information

© Birkhäuser Boston 1991

Authors and Affiliations

  • Ker-I Ko
    • 1
  1. 1.Department of Computer ScienceState University of New York at Stony BrookStony BrookUSA

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