Another brilliant and somewhat unexpected idea described in the article was the idea of variation by way of replacement of the independent variable (time), or v -replacement, where v is a derivative of the function that makes the replacement. Any monotonous time replacement (i.e. replacement with the nonnegative function v) creates the so-called 'attached problem' and a new optimal trajectory in it. Small variations of function in the attached problem lead to nonsmall variations of control in the original problem, which act as Weierstrass needle-shaped variations used to obtain the maximum principle initially.
The local maximum principle described in each attached problem is rewritten as a 'partial maximum principle' in the original problem. The maximum principle results from the organization of partial maximum principles in the original problem (which was possible in all the cases explored at the time, and many other cases as well).
So the maximum principle received a new important interpretation: it turned out to be equivalent to the condition of stationarity in each attached problem (what is meant here is the stationarity of the trajectory which stemmed from the original trajectory by way of v-replacement). Later, this interpretation served as a reliable point of reference in cases where the maximum principle was extended to broader classes of problems.
Problems with state constraints were the first to be covered. The maximum principle for problems with state constraints was the subject of the doctorate thesis A.A. Milyutin defended brilliantly at the Institute of Applied Mathematics (USSR Academy of Sciences) in 1966. In addition to major findings related to the maximum principle the thesis explored an example of an extremal which, when it 'lands' on the boundary of a state constraint, has a countable number of contacts with the boundary on the finite segment of the time preceding the movement along the boundary. The control here has a countable number of switchings which accumulate towards the point (chattering). The moments of the switchings formed a decreasing geometric progression - that was related to the self-similarity of the trajectory. Later a similar example was independently discovered and computed by Robbins.
In the late 1960s and 1970s A.Y. Dubovitsky and A.A. Milyutin jointly developed the theory of the maximum principle for problems with regular and nonregular mixed constraints, which they published in a number of articles. Their extraordinary achievement was the local maximum principle for nonregular mixed constraints, explored in the so-called 'White Book', Necessary Conditions of Extremum in the General Problem of Optimal Control, 1971, Nauka Publishers, Moscow. Unfortunately with the circulation of just 500 copies, the book has long become a bibliographical piece. The book conducts a trenchant analysis of the Eiler equation for problems with mixed constraints, which involves functionals from the space conjugate to L00 , in particular their singular constituents. The analysis made sure that no information was coarsened or lost. The answer was given in the terms of summed functions and Radon measures and by its characteristics it was close to the solution previously obtained for problems with state constraints.
The authors' subsequent efforts were made towards obtaining an integral maximum principle for problems with nonregular mixed constraints. It turned out however that unlike in the case of regular problems, in the general case one can't expect to obtain a single maximum principle, instead one has to deal with a whole hierarchy of maximum principles without the 'maximum element'. So the authors focused on looking for the best possible way to present and organize the hierarchy. That was the subject of A.Y. Dubovitsky's doctorate thesis. Later A.A. Milyutin found a new form of presenting the conditions of the maximum principle; the form reflected multiplicity and hierarchy of maximum principles in the general problem, as well as new ways of obtaining those. That material was presented in A.A. Milyutin's monograph, The Maximum Principle in the General Problem of Optimal Control, 2001, PhizMatLit Publishers, Moscow.
Besides conducting research, in the late 1960s and early 1970s Milyutin started holding lectures and seminars for the DMM students at Moscow State University. It was at the seminar which Milyutin co-led with Y.S. Levitin, that he began in-depth research into the theory of higher order conditions. He raised the issue of obtaining necessary second order conditions in optimal control, which would be connected with the sufficient conditions as closely as in the problems of analysis and calculus of variations. The research resulted in the general theory of higher order conditions in problems with constraints, to which the new concept of condition order was central. Now the order was interpreted as nonnegative functional in variation space, which served as assessment for the problem's functional increment at the point of minimum in admissible variations and determined the degree of the roughness of the condition in question. That theory was published in the article by Y.S Levitin, A.A. Milyutin, N.P. Osmolovsky in Russian Mathematical Surveys, #6, 1978 dedicated to L.S. Pontryagin's 70th birthday. The theory offered landmark approaches to obtaining higher order conditions in optimal control, and the results were ready to follow. The first serious findings by the authors and A.V. Dmitruk were described in the above article; later A.V. Dmitruk and N.P. Osmolovsky, Milyutin's former students, generalized their findings pertaining to the theory of quadratic conditions in optimal control for singular and nonsingular extremals respectively, and defended their doctorate theses on the subject. A.A. Milyutin led their research and took an active part in it. At about the same time he proved the remarkable 'theorem on finite commensurability' which revealed the true meaning of a whole number of findings by other mathematicians (A.A. Agrachev, R.V. Gamkrelidze, Krener and others) in the field of higher order necessary conditions for singular regimes in optimal control.
In those years, mathematicians who focused on the extremum theory, including A.D. Ioffe, V.M. Tikhomirov, V.F. Sukhinin and others, were discovering new forms of the theory's central piece, Lyusternik's theorem on tangent manifold. A.A. Milyutin offered a different interpretation of Lyusternik's theorem and defined it as a 'theorem on covering'. More interpretations were found later. The review of the findings was published by A.V. Dmitruk, A.A. Milyutin and N.P. Osmolovsky in the Russian Math Surveys issue dedicated to L.A. Lyusternik's 80th birthday. But the theorem on covering turned out to be the easiest and most understandable in terms of its formulation and very practical in terms of its use. That's the reason why it's gaining popularity, in nonsmooth analysis among other fields (see A.D. Ioffe's article in Russian Math Surveys, Vol. 55, Issue 3, 2000).
In about mid-80s Milyutin changed focus from obtaining new extremum conditions to making these conditions as much suitable for investigating new phenomena in optimal control as possible. That led to the theorems on the special structure of Lagrangian multipliers in the maximum principle conditions (theorems on the absence of jumps and theorems on the absence of singular components in Lagrangian multiplier measures under state constraints), which were described in the monograph Necessary Condition in Optimal Control, 1990, Nauka Publishers, Moscow. Moreover, Milyutin was thoroughly exploring the phenomena in optimal control and other fields of mathematics using the tools of the maximum principle and higher order conditions.
The maximum principle was used to investigate extremals when they land on the boundary of a state constraint and take off it. The findings were described in the monograph by V.V. Dikusar and A.A. Milyutin Qualitative and Numerical Methods in Maximum Principle, 1989, Nauka Publishers, Moscow. Later, A.A. Milyutin developed an exhaustive solution of a number of special problems related to extremal landing on the state constraint. There he defined the conditions under which the extremal landing was accompanied by a countable number of contacts with the state constraint.