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I think this problem is known in the literature as 'soft landing'. A nice article which may interest you is (1) and the references therein. (1): Liu, Xing-Long, Guang-Ren Duan, and Kok-Lay Teo. "Optimal soft landing control for moon lander." Automatica 44.4 (2008): 1097-1103.
I think you might be interested in peridynamics. It is an approach to continuum mechanics that is based on an integral formulation rather than a differential one where a point in the material is governed by interactions with neighboring points within a given radius. The approach itself is not discrete but the implementation is and most of the works about ...
An axiomatic approach is not as valuable as you think. This is true even in math... Some math textbooks will define the algebra of real 2x2 matrices as "four real numbers in a 2x2 grid with the following addition and multiplication rules", and then later they will prove that matrices are equivalent to the linear operators on a 2D real vector space with a ...
The situation (definitions vs. axiomatic interpretation of laws, theories, etc.) is even less satisfying as you suggest in your question. When Newton gave us his second law, F=ma, these quantities are defined in terms of length, time, mass, and energy, but I would point out that it does so without ever defining what a length or a time actually is. ...
Also, see the lectures by Maciej Dunajski Twistor Theory and Differential Equations (there are also slides available) and his book Solitons, Instantons and Twistors
There is a recent textbook which gives a fairly complete and concise presentation of group theory, covering both structure and representations of both finite and continuous (Lie) groups, with a brief discussion on applications to music (finite groups) and elementary particles (Lie groups). The target level is advanced undergraduate and beginning graduate. It ...
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