Even at the problem statement stage, where besides equations one has to formulate additional conditions, for example initial or boundary conditions or asymptotic at infinity, he was not just guided by the assessment of the validity of the probable theorem on the unique existence of equations with additional conditions, proposed by the physicists based on their knowledge of the process properties, including verified properties, but tried to understand their nitty-gritty. By analyzing the situation and paraphrasing the statement, he would finally come to the correct statement evolving the solution sought by the physicists, as the computation process should allow for physical laws and properties of the phenomenon under exploration; in some instances that leads to the unambiguously most reliable algorithm.
Another example is A.A. Milyutin's approach to the development of the solution method which he referred to as the 'trivialization' of original equations. What was used was not just their simplified model - it was maximum possible simplification which retained the general properties of the original equations, if possible the most material properties. An algorithm was developed for the simplified problem and generalized to the original one. A.A. Milyutin had a special gift for the trivialization which 'did not throw out the child'. His approach was very impressive and often highly effective.
After L.A. Chudov left ICP A.A. Milyutin was appointed as the group leader. Milyutin's group streamlined computational support for ICP laboratories. It was a time when the assignments grew in number and, dramatically, in complexity. It was also a period of transition to computer solution of the whole array of problems. The problems covered a great variety of subjects. They included numerous examples of chemical reaction kinetics, burning and explosion processes, computations related to chemical reactors, gas dynamics with strong shock waves, strong explosions in nonhomogenious atmosphere, electromagnetic radiation transport in the air, computation of volt-ampere characteristics in electrical circuits containing chemical solutions, etc.
It so happened then that Milyutin's group went beyond the Institute's limits up to a national, even international, level. At the time, the USSR and USA sent their representatives to a conference in Geneva to discuss a ban for nuclear weapon tests in all environments. The conference was halting because unlike Soviet experts, the Americans were convinced that nuclear explosion signals could be confidently distinguished from natural phenomena only at a small distance from the venue of a possible breach of the future treaty. So the Americans were asking for permission to deploy radars in the USSR, which the Soviet side couldn't agree to.
Milyutin's group was given an assignment to calculate nuclear explosion effects under two scenarios, in the air and in a spherical cavity in the ground. The "air"-related calculations were tied to a special rf pulse discovered by ICP scientists. The pulse is radiated at the time of a nuclear explosion. The calculation showed that even at long distances, a nuclear explosion can be confidently distinguished from other bursts or explosions.
In the "ground" case, the calculations had to cover a period between the formation of a strong shock wave in the cavity center as a result of a nuclear explosion and the sound stage of the flow; that included numerous wave reflections from the cavity walls and their collapse in the center, a very difficult task for the time. The air impact on the cavity walls throughout this long period of time was included as an additional condition for a problem about geophysical wave propagation along the earth's surface. As a result, it became possible to confidently distinguish explosion characteristics from similar characteristics of earthquakes, even at long distances. When these results were presented to the American experts it turned out that they were not familiar with such reasoning. Some time later, the Americans ran analogous calculations and received the same results. The Treaty was signed without any foreign radars in the Soviet Union.
After considerable expansion of the computational division at ICP, acquisition of a powerful computer and the appointment of A.Y. Povzner as head of the Mathematical Department, A.A. Milyutin focused on pure mathematics. But he did not give up applied problems and continued to be involved in the statement and calculation of specific problems which were important for the Institute.'
One the most brilliant results of the late 1950s, the result which made a real stir and infused life into a new development in mathematics and mathematical applications was L.S. Pontryagin's famous maximum principle. The maximum principle proof obtained by L.S. Pontryagin together with his co-workers V.G. Boltyansky, R.V. Gamkrelidze and Y.F. Michshenko was rather different, in terms of style, than the proofs of extremum necessary conditions well known in analysis and calculus of variations. To some extent, it seemed natural due to the originality and a much higher degree of complexity of optimal control problems as compared to calculus of variations problems. Nevertheless, the issue of the connection between and continuity of calculus of variations and optimal control remained undetermined. This issue stimulated development of new proofs of the maximum principle and new approaches to obtaining necessary first-order conditions in optimal control in this country and abroad. For a while optimal control turned into a real Klondike where everyone tried to find a golden bullion.
The question about whether there was a way to prove the maximum principle using traditional approaches and methods was the one A.Y. Povzner asked of two ICP fellows, A.Y. Dubovitsky and A.A. Milyutin. Quite unexpectedly, the question became a turning-point for both of them and twisted their future work (which they did jointly for a rather long period of time) with optimal control. In 1965 the Computing Mathematics and Mathematical Physics journal published, on N.N. Moiseev's initiative, an article by A.Y. Dubovitsky and A.A. Milyutin, "Extremum Problems under Constraints". The article immediately won popularity and became pivotal for the authors and some of their followers due to its unique clarity and elegance.
Any necessary first-order condition of local minimum in a problem with constraints was interpreted in the article as a condition of nonitersection of the constraints approximation with the approximation of minimized functional decrease set; the set of decrease and all inequality constraints were equivalent and examined independently of each other. First-order approximations for inequalities (under natural assumptions) are open convex cones while equality constraints approximation is just an open cone (as a rule, it's a subspace according to L.A. Lyusternik's theorem on tangent manifold). The condition of nonintersection of cones, which the authors called the Eiler equation, is exactly the necessary first-order condition. When defined, the condition needs to be explained in the language of the field of mathematics where the problem is explored. In optimal control, the condition of cone nonintersection leads to the condition which is typically referred to as the Eiler-Lagrange condition, or according to the authors, the 'local maximum principle".