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In a recent paper, Seth has pointed out the error in the derivation of the adjoint model of Director and Rohrer and has given the correct results which are applicable for time-domain analysis. This fact calls for a re-examination of the results obtained from considerations of inter-reciprocity between the sensitivity network and the adjoint network. The purpose of this paper is first to develop the sensitivity model in its most general parametric form and then establish a link with the corrected adjoint model through the use of the inter-reciprocity theorem.

In this paper it is proved that A-integrals and B-integrals are inconsistent with the Denjoy-Hincin integral of functions which are adjoint to a summable function. Furthermore, the paper establishes nonadditivity of the B-integrals on an interval.

The results stated in this chapter were obtained by the Author in his paper [2] and substantially extended in the paper by N. I. Muskhelishvili and D. A. Kveselava [1]. In this latter paper the concept of classes of solutions was introduced for the first time and the fundamental theorems of § 112 were proved. In addition, another method of investigation due to D. A. Kveselava [1] is given in § 115.

THIS PAPER will report on the investigation of certain problems in the optimum control of a linear dynamic system, particularly the problem of determining the minimum time required to drive a linear, constant coefficient dynamic system from an initial state to a specified terminal state with a limited power source. An important feature of this treatment is that an elementary method of solution for the problem is given. This is a method of successive approximations based on the adjoint system of differential equations. The method is similar to that which Bliss used in calculating differentials in Ballistics. A program for solving the minimum-time problem on a digital computer will be presented, and elementary methods will be used to prove that if the routine converges, then the solution thus found yields the desired minimum time.

This paper contains a proof that either all, or none, of the nontrivial solutions of the fourth order linear selfadjoint differential equation have an infinite number of zeros on a half line, provided that no nontrivial solution has more than one double zero on that half line. Throughout this paper, let Ly = (ry")" -(qy')'+py where r, q, and p are given real-valued functions, a?(o, oo) is given, r", q', pCC[a, oo), and r(t)>O for t_?a. A nontrivial solution to Ly=O is said to oscillate if its zeros in [a, oo) are unbounded. THEOREM 1. If no nontrivial solution to Ly = 0 has more than one double zero in [a, oo), then all the nontrivial solutions oscillate or none oscillate. W. Leighton and Z. Nehari [1, p. 367 ] obtain the same conclusion using the hypothesis that q(t) 0, p(t) > 0 for t > a. As they note, these assumptions imply that no nontrivial solution has more than one double zero. Some lemmas will be established before proving Theorem 1. Ly = 0 will be said to be 2-2 disconj ugate if no nontrivial solution has more than one double zero in [a, oo). Of course, this is a special case of the concept known as n-n disconjugacy. When r, q, and p are all constants, it can easily be shown that Ly =0 is 2-2 disconjugate if and only if rw4+qw2+p > 0 for all real numbers, w. For i = 1, 2, 3 and a < b < oo, let ybi designate the solution to Ly = 0, y(i)(b) = bij, j = 0, 1, 2, 3, where 6ij is the Kronecker delta. Denote the zeros of Yb3 by ,[ . . < -q(b .1 < nb 1)

One first finds the Riemann function, which is the solution of a homogeneous adjoint equation subject to simple boundary conditions that are independent of the given boundary data. The solution of (1.1), for any appropriate boundary data, can then be obtained by evaluating a definite integral, where the Riemann function and the boundary data appear in the integrand. The main obstacle to the use of this method for numerical computation has been the difficulty in finding the Riemann function. It was pointed out in a 1947 paper by Cohn [2] that, aside from a(xi, X2) = constant and a(xi, X2) = k(k 1) * (xI + x2f2, which was treated by Riemann, there are very few cases where expressions for the Riemann function have been obtained. One can, of course, use the Picard method of successive approximations, but this is usually not practical. In this paper we derive a simple recurrence formula for the Riemann function, for the case where a(xi, X2) can be expanded in a Taylor series about the point where the solution is sought. We consider here the Riemann function for the operator L in the n dimensional analogue of (1.1), (1.2) L(u) = CnU/C1XOX2 ... **xn + a(xi, x2, X * *, )u = F (xi, x2, Xn)

This paper presents a practical formulation for the solution of the elastostatic problem of a semi-infinite solid cylinder with the long sides free from stress and self-equilibrated tractions applied on the end. The formulation is entirely in terms of stresses and displacement related auxiliary variables of the same differential order as the stresses. The equilibrium equations, the Beltrami-Michell equations of compatibility and the definitions of the auxiliary variables are used to write a first order matrix differential equation. The solution of this non-self adjoint equation yields a series of non-orthogonal vector eigenfunctions. The coefficients of this series are chosen by use of a generalized biorthogonality condition. Numerical solutions of trial problems are presented as an indication of the rate of convergence of this series.

1. That adjoint differential equations have an analogue in the theory of linear difference equations seems to have been first observed by Bortolotti. The relation is essentially that of a matrix ǁ a rs ǁ to its transposed matrix ǁ a sr ǁ. It seems desirable, from this point of view, to carry out the transition from difference to differential equations, and thus prove that the analogy is a real one. This is done in Art. 2. There are further consequences of general interest. A set of linear equations corresponds to a differential equation and its boundary conditions, and thus we can find an interpretation of the adjoint boundary conditions introduced by Birkhoff into the theory of linear differential equations (Arts. 3-6). The relation between the two Green’s functions, implicit in Birkhoff’s work, then becomes evident (Art. 7). 2. We first prove that if the equations a r 1 y 1 + a r 2 y 2 + ... + a rn y n = fr ( r = 1 to n ) (1) are so constituted that they merge into the differential equation L ( y ) Ξ a m d m y / dx m . . . + a 1 dy / dx + a o y = f (2) by passing to an infinite number of infinitesimally spaced unknowns, the transposed equations a 1 r z 1 + a 2 r z 2 + ... + a nr z n = g r (3) merge into the adjoint equation M ( z ) Ξ (—) m d m / dx m ( a m z ) + ... - d / dx ( a 1 z ) + a o z = g .(4)

The paper presents a topological correspondence between a network and its adjoint which, in the final result, renders it possible to proceed to the adjoint network without going through the intermediate step of deriving the co-state equations. Algorithms are developed for arriving at the topology of adjoint networks for the terminal control problem of a linear network with a linear criterion as well as with a quadratic criterion. The results are also extended to non-linear networks. The topological interpretation of adjoint networks facilitates inclusion of the results into presently available programmes on computer-aided design, which, in fact, provides the primary motivation for the work.

The adjoint network concept, originally developed for the efficient and accurate computation of the gradient of network performance in parameter space, is shown to be of great utility in other circuit applications. Some of these, including the efficient computation of the sensitivities either of a desired output variable or of the poles and zeros of a network function, the calculation of the group delay, the estimation of the effects of parasitic dissipation and how to compensate for them, the computation of the natural frequencies and a technique for fast statistical tolerance analysis, are summarized in this paper. It is also shown that some specialized network optimization algorithms, believed to be faster and more efficient than either the Fletcher-Powell or the conjugate gradient method, can be based entirely on equations derived from adjoint network considerations.

A method is presented for the systematic evaluation of two-point boundary value problems. Emphasis is placed on solving trajectory optimization problems formulated by the calculus of variations. The method presented in this paper, termed the adjoint method, iteratively converts the two-point boundary value problem into an initial value one, thus allowing a solution to be achieved in one run on a digital computer. Two sample problems that were solved by using the method are given: the case of a boost mission and that of an orbit transfer mission, both to be done in a minimum amount of time.

The synthesis of optimum controllers is a topic of considerable current interest. Whereas many solutions to problems of optimization are available there are relatively few proposals in the literature for closed-loop, optimum system construction. This paper proposes that (a) the constancy property, associated with the Pontriagin Hamiltonian for linear, time-invariant plants, be used to determine the initial conditions necessary for the adjoint equation with time-optimum problems and (b) proposes a closed-loop configuration that utilizes this approach. The operation of the proposed system is discussed, its limitations arc stated and the stability is indicated in a special case.

This paper describes tlie algorithms for effective calculation of statistical characteristics of linear nonstationary composite systems. First the adjoint equation method for calculation of weight functions is developed for such systems with nonhomo-geneous means of description (vectors and matrices of different dimension, different types of equations etc.), in particular for the systems with continuous and discrete elements. Then the exact structure of the shaping filter method and the new forming equation method axe suggested for nonstationary regimes. By means of these methods a covariation matrix of the system can be determined by simultaneous integration of certain equations with weight matrix equations.The different procedures are formed to determine covariation matrix for fixed and running time.

is an extension of m. Recall that X, Y, Z are naturally embeddable in X-, Y--, Z-resp. Moreover, certain properties, such as associativity, when m has them, are transmitted to m*** (this makes sense only when Y=Z=X). On the other hand, the transmission of commutativity (which makes sense when Y=X) was left open, and will be considered in this paper. This question of commutativity can be generalized as follows. If m satisfies 1.1-1.3, one can define the transposed operation

An important class of optimal control problems, arising frequeutly in an economic framework, is characterized as having a cost functional that is continuous but has discontinuous partial derivatives with respect to the state variables. Such problems are said to have kinks. Along a kink the classical adjoint equation breaks down, and it is impossible to define a gradient. In this paper it is shown that the gradient can be replaced by a more general definition of the direction of steepest descent but that the adjoint equation must in general be replaced by an adjoint optimal control problem. This yields a complete set of necessary conditions for problems of this type. The results derived are then combined with the theory of penalty functions to convert a problem having state constraints to one without such constraints.

then / is identically zero. Because of this result .{] : / C Lp (~ , + oo), / ~nd [' absolutely c6ntinuous and ]" in Lp ( ~ , -7 ~)} is the saturation class (see [8], p. 287) for the approximat ion method {Tt}. The basic theorem from which most of his results follow is valid only for reflexive Banach spaces and so does not yield an identification of the saturat ion classes for the extreme values p = t and p = ~ . In this paper we identify some of these saturat ion classes by consideration of the appropriate adjoint semigroup. The following section is devoted to the necessary general semigroup results. These results are applied in Section 3 to obtain some identifications of Lipschitz classes due to HARDY and LITTLEWOOD, and in the remaining sections are applied to identify saturat ion classes.

The problem of the slowing down of neutrons in an infinite homogeneous medium with strong resonance absorption and uniformly distributed neutron sources is investigated in this paper. The solution of the adjoint equation represents the probability that a neutron of energy E escapes resonance absorption during the process of slowing down to a certain asymptotic energy. The solution of the main and the adjoint problems makes it possible for us to apply a perturbation method to take into account the influence on the resonance integral of the Doppler broadening of the resonance level. The methods developed have been applied to the calculation of the collision density and the resonance integrals for the first level of U238 (E0 = 6.7 ev) in pure uranium and in uranium oxide UO2.

This paper shows how the basic idea of the adjoint method can be taught to juniors using only sinusoidal steady-state analysis, Cramer's rule, a few matrix transpositions and the relation between node-admittance matrix and branch-admittance matrix.

In the first part of this paper we have derived an adjoint system of equations for the set of equations characterising the solution of the stability of viscous flow between two rotating cylinders, when the marginal stability is assumed not to be stationary. Then the adjoint system of differential equations has been solved to arrive at a simpler secular equation than the one obtained by Chandrasekhar. By a different approach than that of Chandrasekhar's, an attempt is made to show that for μ, which is defined as the ratio of the velocities Ω1 and Ω2 with which the inner and outer cylinders are rotated, greater than zero, there is no possibility of the instability setting in as overstability.

The algebraic relation between state and adjoint variable initial conditions for time-optimum control of a particular second order linear system is reported. The relation between adjoint initial conditions and switching time suggests that difficulties might arise when boundary-value search methods are employed to solve two-point boundary-value problems. These analytical results support the conclusions in a recent paper by Paynter and Bankoff.

Summary This paper gives an expository discussion of the Pontryagin Maximum Principle, comparing it to the more classical variational techniques. In the Pontriagin Approach we see that the auxiliary p variables are the adjoint system variables. Further, we find that the Maximum Principle can be derived from an extension of the properties of adjoint systems, which is motivated by one of the well-known linear properties of adjoint systems. The equivalence of the Weierstrass Condition and the Maximum Principle has been realized by many people for the case of unbounded controls. By adjoining the inequality constraints via the Valentine technique, we find the Weierstrass Condition remains valid when properly interpreted. Publisher Summary This chapter focuses on the Pontryagin maximum principle. In the Pontriagin approach, the auxiliary p variables are the adjoint system variables. The maximum principle is derived from an extension of the properties of adjoint systems that is motivated by one of the well-known linear properties of adjoint systems. Maximum principle is equivalent to the Weierstrass condition when no constraints are imposed on the control variables. In the preceding development of the maximum principle, it is assumed that final time tf is fixed. For final time opened, it is shown that the additional relationship needed is that the Hamiltonian H is zero at tf. Constraints on the final values of the state variables are included using the Lagrangian multipliers. This changes the boundary conditions on the auxiliary p variables.

Kolmogorov [13] has shown that under appropriate regularity conditions such a P(t) will satisfy a pair of matrix differential equations (adjoint to each other). Since then a great many investigations have been concerned with problems of non-uniqueness, with solutions which fail to satisfy the adjoint equation, and with other "pathological" situations. It is the purpose of the present paper to show that the unbounded matrix operator Q on which the Kolmogorov equations depend shares the essential properties of second order elliptic differential operators, such as the SturmLiouville operator in one dimension or the Laplacian A in two dimensions. The Kolmogorov equations are then the exact counterpart of the heat equation ut = Au, except that due to the essential lack of symmetry of Q the adjoint equation bears little resemblance to the original. Our solutions will depend on boundary conditions in which one easily recognizes the classical boundary conditions of, say, the theory of harmonic functions. Of course, the latter depend on normal derivatives and no analogue to partial derivatives exists in our case. However, our boundary conditions are expressed in terms of functionals which remain meaningful under the most general circumstances, and can be applied to harmonic functions. There they reduce to the normal derivatives whenever the boundary is sufficiently smooth, but they give the proper expression for an arbitrary boundary. The perfect analogy of the Kolmogorov equations with diffusion equations explains the many phenomena and problems connected with the non-uniqueness of the solutions, and leads to new insights. Consider, for simplicity, an isolated point X of the boundary. It is possible to extend Q (that is, the Kolmogorov equations) and the matrix P(t) to the countable set E + a. This extension is not uniquely determined, but it involves very little arbitrariness. It adds a new row and a new column to Q, and these bear no similarity to the remaining rows and columns of Q. This leads in the most natural manner to generalizations of the