On variational problems of optimization of control processes

Troitskii, V.A.

doi:10.1016/0021-8928(62)90100-4

Cited by 9 publications

(3 citation statements)

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“…Many authors emphasize close connections between the calculus of variations, the max/min principle, and dynamic programming, see [8, p. 91] and also [13][14][15]. Consequently, the invalidity of dynamic programming for certain classes of optimal problems has implications for the two preceding general methods in control.…”

Section: Discussionmentioning

confidence: 99%

“…A careful reader can see that it is the equalities of type (7.10), (7.11) that are actually necessary and sufficient for the derivation of the functional equations and PDEs of dynamic programming in [1,Ch. 9,Sections 3,6,13,22], as it is presented above in (7.1) to (7.9). Denote S --t2 -tl (same S as in [1, Ch.…”

Section: X[t:t] = Lim (X[t:t2] U X*[t2t])mentioning

confidence: 90%

“…Remark: if those PDEs are contradictory, or do not define an optimal control, or other hurdles occur, see Example 6.1, this does not mean nonexistence of optimal solutions. In such a case, try other methods, e.g., [6][7][8][9][10][12][13][14]. STEP 6.…”

Section: Step 3 Ifmentioning

confidence: 99%

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Reflections on optimality and dynamic programming

Galperin

2006

Computers & Mathematics with Applications

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Discrete models and continuous control systems are considered in regard to optimality of their trajectories. Some aspects of the principle of optimality [1, p. 83] axe analyzed, and it is shown to imply total optimality, that is, the optimality of every part of an optimal trajectory. Certain autonomous systems with free admissible variations possess this property.Nonautonomous optimal systems axe not, in general, totally optimal, in which case the principle of optimality is not valid. A modification is proposed for the derivation of the main functional equation to demonstrate that dynamic programming and its functional equations are valid also in the case of nonoptimal remaining trajectories under a certain contiguity condition that is defined and analyzed in the paper.Control systems with incomplete information or structural limitations on controls do not, in general, satisfy the contiguity condition. Control problems for such systems may have optimal solutions which, however, cannot be obtained by dynamic programming. This fact is shown in an example of a widely used engineering system for which an optimal trajectory has all its remaining parts nonoptimal and noncontiguous to the optimal trajectory. The paper presents theoretical justification of dynamic programming for contiguous systems that do not conform to the principle of optimality. Examples axe presented to illustrate the results which open new avenues in modeling and optimization of general (not totally optimal) control systems. (~)

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Section: Discussionmentioning

confidence: 99%

Section: X[t:t] = Lim (X[t:t2] U X*[t2t])mentioning

confidence: 90%