Simultaneously, the maximum principle was used to investigate the behavior of extremals during their passage through a singular manifold (in particular, during the landing on a singular extremal), the nonuniqueness funnel for extremals, the conditions for the chattering-type effect or, more generally, discontinuity of the second kind of control. S.V. Chukanov took an active part in the research. The results were published in Optimal Control in Linear Systems by A.A. Milyutin, A.E. Ilyutovich, N.P. Osmolovsky, and S.V. Chukanov.
A.A. Milyutin used quadratic conditions to investigate the concept of rigidity in optimal control. The complete system of quadratic conditions is transferred to rigidity. That results in obtaining characteristics of sets of quadratically rigid trajectories in contact structures for the arbitrary dimension space and determining the structure and dimension of the quadratically rigid trajectory set in a contact structure depending on the space dimension. Milyutin built a seemingly surprising example of a quadratically rigid trajectory with a nonunique set of normed Lagrangian multipliers where the quadratic form corresponding to each set is not even negatively defined in a set of critical variations (the example is no longer surprising when one gets to know the complete system of quadratic conditions for singular extremals obtained by A.V. Dmitruk).
Together with A.V. Dmitruk, A.A. Milyutin used quadratic conditions to analyze singular geodesics relative to submetrics. In particular, he established nonuniformity (in terms of extremal characteristics) of the vicinity structure of a singular geodesic relative to a submetric and proved that in the topology of uniform convergence of functions with their derivatives, each locally quadratically rigid geodesic is ultimate for the sequences of two types of nonsingular extremals: the sequences where all couples of conjugate points converge at an arbitrary small distance, and the sequences which do not have subsequences of converging couples of conjugate points.
The problems arising in the quadratic theory of singular extremals triggered research into the possibility of approximation of random vector fields in a finite-dimensional space with gradient fields. Milyutin found a duality formula which connects normed vector field circulation with a distance (in a sense) between that field and a set of gradient vector fields. Later, the formula was effectively used in the quadratic theory of singular extremals.
Together with V.L. Bodneva, A.A. Milyutin generalized the Bogolyubov-Krylov asymptotic method for the case of the differential equation right side's random dependence on the parameter. The parameter here may be an element of the metric space. Recursions of the generalized asymptotic process were obtained (Russian Math Surveys, 1987, vol.42, N 3). That approach helped the authors (Milyutin and Bodneva) obtain new interesting results related to the mathematical theory of vibrations, where vibration, a sequence weakly L00 converging to zero, is viewed as a small parameter. (Russian J. Of Math. Phys., 1998, vol.5, N2). The proposed method goes beyond the known averaging methods and helps obtain ultimate systems in the case of right sides which are continuous in a state constraint and in the case of discontinuous right sides as well (an example being problems involving dry friction).
Together with N.P. Osmolovsky, A.A. Milyutin explored the movement and interpenetration of ideas between calculus of variations and optimal control. Milyutin started this exploration with a course of lectures on calculus of variations at the DMM of Moscow State University. By analogy with the weak minimum theory which makes the foundation of the classical calculus of variations, what is built is the calculus of the so-called Pontryagin minimum explored in the optimal control theory. The Pontryagin minimum theory was described in a monograph by A.A. Milyutin and N.P. Osmolovsky, Calculus of Variations and Optimal Control, published by the American Mathematical Society in their Translations of Mathematical Monographs series in 1998.
After his trip to Israel in 1998 where he attended a conference dedicated to the tercentenary of calculus of variations, Milyutin focused on the findings in the field of the maximum principle theory for differential inclusions. He explored the influence of the Holder condition for a differential inclusion on the form of the maximum principle and established that for a differential inclusion whose right side is prescribed by convex-valued mapping which meets the Holder condition, in the absence of state and mixed constraints there is no maximum principle with continuous conjugate variables (which gives a partial answer to Clarke's similar question about the Lipschitz inclusions). For differential inclusions, Milyutin also established the nonequivalence of the maximum principle to the Ecklund principle. Using the maximum principle in optimization problems for differential inclusions he identified finer necessary conditions that those obtained earlier by Clarke who used the Ecklund principle in the same problems. He also used the maximum principle to enhance the Smirnov conditions.
Unfortunately this is too short a review to mention all important and numerous results obtained by A.A. Milyutin.