We propose the basis for a rigorous approach to modeling combat, specifically under conditions of complexity and uncertainty. The proposed basis is a tensorial generalization of earlier Lanchester-type equations, inspired by the contemporary debate in defence and military circles around how to best utilize information and communications systems in military operations, including the distributed C4ISR system (Command, Control, Communications, Computing, Intelligence, Surveillance and Reconnaissance). Despite attracting considerable interest and spawning several efforts to develop sound theoretical frameworks for informing force design decision-making, the development of good frameworks for analytically modeling combat remains anything but decided. Using a simple combat scenario, we first develop a tensor generalization of the Lanchester square law, and then extend it to also include the Lanchester linear law, which represents the effect of suppressive fire. We also add on-off control inputs, and discuss the results of a simple simulation of the final model using our small scenario.

In the first paper of the tensor-centric warfare (TCW) series [1], we proposed a tensor model of combat generalizing earlier Lanchester-type systems with a particular emphasis on contemporary military thinking, including the distributed C4ISR system (Command, Control, Communications, Computing, Intelligence, Surveillance and Reconnaissance). In the present paper, we extend this initial tensor combat model with entropic Lie-derivative machinery in order to capture some aspects of this deep uncertainty, while, in the process, formalizing into our model military notion of symmetry and asymmetry in warfare as a commutator, also known as a Lie bracket. In doing so, we have sought to shift the question from the prediction of outcomes of combat, upon which previous combat models such as the Lanchester-type equations have been typically constructed, towards determining the uncertainty outcomes, using a rigorous analytical basis.

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This paper is the third part of the complex combat dynamics series, called tensor-centric warfare (for the first two parts, see [1] [2]). In the present paper, we extend the tensor combat model from [1] and [2] to model the dynamics of delta-strikes/missiles, which are temporally confined strong kinetic effects. The scenarios analyzed here include both deterministic and random delta-strikes which mimic single, multiple and continuous-time missile attacks. We also look at the bidirectional random strike as well as the general Hamilton-Langevin dynamics framework and provide an interpretation of the results obtained through simulation.