Abstract
Necessary conditions are proved for multi-dimensional control systems, in which the state is a function of several independent variables in a fixed domain, while the controlu is a function of only some of the independent variables (that is,u is a lower-dimensional control). The cost functional can be taken in the usual Lagrange form of a multiple integral or in Mayer form. The state equations are a system of partial differential equations in explicit form with assigned boundary conditions, and constraints may be imposed on the values of the control variables.
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References
Baum, R.,Existence Theorems for Multidimensional Control Systems with Lower Dimensional Controls, SIAM Journal on Control, Vol. 10, pp. 623–638, 1972.
Cesari, L.,Optimization with Partial Differential Equations in Dieudonne-Rashevsky Form and Conjugate Problems, Archive for Rational Mechanics and Analysis, Vol. 33, pp. 339–357, 1969.
Suryanarayana, M. B.,Necessary Conditions for Optimization Problems with Hyperbolic Partial Differential Equations, SIAM Journal on Control, Vol. 11, pp. 130–147, 1973.
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Communicated by L. Cesari
The author wishes to thank Professor L. Cesari for his many helpful comments and assistance in the preparation of this paper and Professor L. DeBoer for his interest and encouragement. This work was sponsored by the United States Air Force under Grants Nos. AF-AFOSR-69-1767-A and AFOSR-69-1662.
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Baum, R.F. Necessary conditions for distributed-parameter systems with controls of fewer variables than state variables. J Optim Theory Appl 30, 663–681 (1980). https://doi.org/10.1007/BF01686728
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DOI: https://doi.org/10.1007/BF01686728