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1971

Volume 9, Issue 4, pp. 519-633


Weak Solutions of a Partial Differential Equation of Dynamic Programming

Raymond W. Rishel

SIAM J. Control 9, pp. 519-528 (10 pages) | Cited 2 times

Online Publication Date: July 18, 2006

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A formal dynamic programming argument relates the “value function of a stochastic optimal control problem” with the solution of a nonlinear parabolic partial differential equation. In cases in which the partial differential equation is degenerate, it may not have a classical solution but may have a weak solution in the sense of the theory of Schwartz distributions. It has been an open question as to whether a weak solution of the partial differential equation does equal the value function of the stochastic optimal control problem.
This paper shows that, roughly, whenever an associated uncontrolled system has an appropriately behaved density function, the equality holds. It is also shown that an optimal control law may be determined by minimizing a Hamiltonian formed in terms of the partial derivatives of the weak solution of the partial differential equation.

Generalized Cutting Plane Algorithms

B. Curtis Eaves and W. I. Zangwill

SIAM J. Control 9, pp. 529-542 (14 pages) | Cited 33 times

Online Publication Date: July 18, 2006

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This paper introduces a master cutting plane algorithm for nonlinear programming that isolates the points it generates from one another until a solution is achieved. The master algorithm provides a foundation for the study of cutting plane algorithms and directs the way for development of procedures which permit deletion of old cuts.

A Note on Complete Controllability

Inge Troch

SIAM J. Control 9, pp. 543-546 (4 pages)

Online Publication Date: July 18, 2006

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For linear autonomous multivariable control systems the conditions are studied under which the system is always—that is, for every control matrix of appropriate rank—completely controllable, which means that for every initial state one can construct a control which steers the system to any given final state in finite time. The resultant necessary and sufficient condition says that the number of independent controls either must equal the system’s dimension or must be smaller by one. In the latter case no eigenvalue of the system matrix may be real. These results are valid for continuous systems as well as for sampled-data systems and are derived by means of well-known criteria for complete controllability and some theorems of matrix theory. Analogous results which are valid for the question of complete observability are formulated too. As a special case, previous results of Hsin Chu are covered.

Abstract Models for the Synthesis of Optimization Algorithms

Gerard G. L. Meyer and E. Polak

SIAM J. Control 9, pp. 547-560 (14 pages) | Cited 3 times

Online Publication Date: July 18, 2006

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In this paper, we present a systematic approach to the problem of synthesis of optimization algorithms. First, we develop abstract models for algorithms. These models guide the inventive process toward “conceptual” algorithms, namely, algorithms which may consist of operations that are inadmissible in a practical method (for example, a conceptual algorithm may require us to find the limit of an infinite sequence at each iteration). Once the abstract models are established, we present a set of methods for converting “conceptual” algorithms falling into the class defined by the abstract models, into "implementable" iterative procedures.

A Generalized Lagrange Multiplier Rule for Equality Constraints in Normed Linear Spaces

D. O. Norris

SIAM J. Control 9, pp. 561-567 (7 pages)

Online Publication Date: July 18, 2006

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It is shown that a necessary and sufficient condition for a differentiable function to have a critical point on an affine constraint set is that a Lagrange multiplier rule hold. The rule applies to problems which are not covered by the standard multiplier theorem. The results are extended to the determination of necessary conditions for a convex function to have a minimum on a nonaffine constraint set.

Justification of the Describing Function Method

A. R. Bergen and R. L. Franks

SIAM J. Control 9, pp. 568-589 (22 pages) | Cited 19 times

Online Publication Date: July 18, 2006

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Explicit conditions are given under which the use of the describing function method to investigate the nature of oscillations in autonomous nonlinear feedback systems is justified. When these conditions are satisfied, bounds are given for the frequency, fundamental magnitude, and higher harmonics of the oscillation based on the describing function approximation.
The feedback systems considered are those which can be decomposed into a linear time-invariant subsystem, not necessarily causal, stable, or finite-dimensional, and a nonlinear frequency independent subsystem, possibly containing hysteresis.
The approach taken is to consider the describing function equation as an approximation to a determining equation for periodic solutions of the autonomous system’s operator equation, and to use local degree theory to guarantee the existence of a solution to the determining equation.

An Existence Theorem for Lagrange Problems with Unbounded Controls and a Slender Set of Exceptional Points

L. Cesari, J. R. La Palm, and D. A. Sanchez

SIAM J. Control 9, pp. 590-605 (16 pages) | Cited 3 times

Online Publication Date: July 18, 2006

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In a number of existence theorems for Lagrange problems with unbounded controls certain growth conditions are required, which generally can be omitted for bounded controls. In the present paper the authors show that even for unbounded controls the growth conditions can be dispensed with in an arbitrary subset in the $tx$-space $E_{1 + n} $, provided such a subset is “slender” according to suitable definitions. Any set contained in finitely many or countably many smooth curves $x = \phi (t)$, $a \leqq t \leqq b$, is certainly slender. Many examples are given.

Nonlinear Programming in Complex Space: Necessary Conditions

Robert A. Abrams and Adi Ben-Israel

SIAM J. Control 9, pp. 606-620 (15 pages) | Cited 17 times

Online Publication Date: July 18, 2006

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Necessary conditions of the Kuhn–Tucker type are given for two classes of nonlinear programming problems over polyhedral cones in finite-dimensional complex space. The first class consists of problems of the form \[{\text{Minimize }} \operatorname{Re} \quad f(z)\quad {\text{subject to}}\, g(z) \in S,\] where $S$ is a polyhedral cone in $C^m $ and $f:C^n \to C$,$g:C^n \to C^m $ are analytic functions. A necessary condition for a feasible point $z^0 $ to be optimal is that there exist a vector $u \in S^ * $ such that $\overline {\nabla f(z^0 )} = [D_z^H g(z^0 )]u$ and $\operatorname{Re} (g(z^0 ),u) = 0$. The second class consists of problems of the form \[{\text{Minimize }} \operatorname{Re} \quad f(z,\bar z)\quad {\text{subject to }} g(z,\bar z) \in S,\] where $f:C^{2n} \to C$, $g:C^{2n} \to C^m $ are analytic. A necessary condition for a feasible point $z^0 $ to be optimal is that there exist $u \in S^ * $ such that $\overline {\nabla _z f(z^0 ,\overline {z^0 } )} + \nabla _{\overline z } f(z^0 ,\overline {z^0 } ) = [D_z^H g(z^0 ,\overline {z^0 } )]u + [D_{\overline z }^T g(z^0 ,\overline {z^0 } )]\overline u $ and $\operatorname{Re} (g(z^0 ,\overline {z^0 } ),u) = 0$. The derivation of necessary conditions for problems of the first class is analogous to that used in the real case. For problems of the second class, necessary conditions are obtained by considering an equivalent problem in the form of the first class.

Representation of Martingales, Quadratic Variation and Applications

E. Wong

SIAM J. Control 9, pp. 621-633 (13 pages) | Cited 9 times

Online Publication Date: July 18, 2006

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In this paper, we present two related results. First, we shall obtain a sufficient condition under which a second order sample-continuous martingale can be represented as a stochastic integral in terms of a Brownian motion. Secondly, we shall show that if $X$ and $Y$ are sample-continuous local martingales (not necessarily with respect to the same family of ($\sigma $-algebras) and if either $X + Y$ or $X - Y$ is almost surely of bounded variation, then the quadratic variations of the two martingales are equal. This rather simple result has some surprising consequences.
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