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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.

Since at least the time of the Military Enlightenment, military organizations have invested considerable effort into developing theories of war and battle. The purpose of such theories is ultimately to inform decisions: what equipment to acquire, what processes to institute, what training and education to develop, what research to conduct, how to conduct operations, and what operations to conduct in the first place. Perhaps unsurprisingly, many such efforts at the development of theories of war and battle have been oriented around the idea of achieving something like a complete and correct theory by which future outcomes might be predicted and thereby the means of determining the means to guarantee, or at least maximize the chances of, obtaining the outcome one desires. This has remained a dominant theme in military thinking ever since the foundational works of early theorists such as Jomini [

Yet as predicted even from the outset by Clausewitz [

Against a broad backdrop where modern military organizations have sought to move away from approaching force development and employment from the point of view of individual platforms as collections of functions, theories such as NCW proved influential because they postulated that the ability of military organizations to collect and disseminate and process data while denying to adversaries the same ability is the crucial determiner of the outcome. The problem, however, is that NCW presented a discredited though intuitively appealing explanation for exactly how this is to occur, with practical consequences that have proven problematic [

These observations may be seen in terms of the limitations of classical systems engineering achievements in dealing with wicked problems [

In defence operation research community it is well known that classical Lanchester-Osipov combat equations (also called Lanchester-style mass action models [^{1} include two forces, Red/Attacker’s strength: R = R ( t ) : ℝ → ℝ and Blue/Defender’s strength: B = B ( t ) : ℝ → ℝ , with their respective initial sizes R 0 and B 0 and their corresponding combat-effectiveness coefficients k R and k B . Lanchester equations are of the following two basic types:^{2}

1) Lanchester square law for direct-aimed fire:

R ˙ = − k R B , ( with R ( 0 ) = R 0 , k R > 0 ) , B ˙ = − k B R , ( with B ( 0 ) = B 0 , k B > 0 ) , (2)

where overdot denotes time derivative, and k R and k B denote individual combat-rate coefficients for the Red and Blue forces, respectively (e.g., tank versus tank, concentration of fire).

2) Lanchester linear law for area―unaimed fire:

R ˙ = − k B R B R , ( R ( 0 ) = R 0 , k B R > 0 ) , B ˙ = − k R B R B , ( B ( 0 ) = B 0 , k R B > 0 ) , (3)

where k B R and k R B denote mixed combat-rate coefficients for Red and Blue forces (e.g., artillery barraging an area without precise knowledge of target locations).

Although similar Lanchester-type models had been extensively used in the past Century, today they are widely regarded as grossly oversimplified representations of modern warfare, at best. This motivates our effort towards a modeling methodology of much higher complexity, including both continuous and discrete spatiotemporal dynamics, as proposed in the present paper^{3}.

For the last two decades, modern defence forces have generally been investing considerable effort in shifting the basis for decision-making for force development, employment and conduct of operations beyond individuation around platforms. Regarding ships, aircraft, armored vehicles and soldiers, for instance, as the base atomic units of military forces that are then packaged hierarchically yields separate Command and Control (C2) channels for different military functions, which are then attached to acicular organizations that necessarily centralism planning and coordination to achieve desired effects. The emergence of modern communications and information technology was consequently broadly seen as offering the potential to dissolve such crystalline arrangements in favor of military forces able to fluidly self-organize in rapidly changing situations to both counter threats and take advantage of opportunities, by enabling collaboration directly between elements formerly widely separated by hierarchy.

Other advances including those the fields of telecommunications, robotics, artificial intelligence and autonomous systems have further opened apparent opportunities for collaborative planning, coordination and rapid response; at its core, modern defence thinking seeks to achieve highly distributed C2 arrangements enabled by communications and information systems in which information can be rapidly disseminated while also being protected from outside interdiction and interference. The extreme instantiation of this lies in the idea that by the provision of such a system, together with the training and procedures to utilize it, “information superiority”―the ability to acquire, transport and process more information than the opposition―will deliver superior ability to apply the effects of military force, and thus at least maximum chances of winning, if not virtually guaranteed complete battlefield domination. The ability for foreseeable battlefield communication systems to provide an infrastructure sufficient to realism the required connectivity, the flawed nature of at least this kind of extreme account has been made manifest by the fact that apparently overwhelming forces, enjoying the full benefits of the best technology has to offer, can and do continue to lose to ostensibly backwards and inferior forces.

While there remains broad agreement that technological developments offer considerable opportunity, the issue of exactly what benefits are implied and how to best utilize the technologies to best effect have remained unsettled. Theories of war and battle intended as explanatory and predictive bases for guiding force development decision-making have been proposed―NCW being an especially prominent example in Western defence departments in the first decade of the 21st Century―but none has proven satisfactory. In the case of NCW, foundational problems with its explanation of methodology with far-reaching practical consequences in the design of systems and over-estimation of the relative benefits of the technology [

Recent years have seen a growing knowledge about, and interest in, the burgeoning knowledge across the sciences about complexity and uncertainty, among defence and military thinkers. The general emerging view is that defence and military matters feature burgeoning complexity of technological, social, economic, cultural and political varieties. Whether war and battle is really becoming more complex than in the past is highly debatable; what is more certain is that analytical and conceptual frameworks used in its study to inform force design decisions struggle to adequately account for effects that Clausewitz pondered 180 years ago. The objective of the present paper is to take a step in this direction by generalizing and making rigorous the study of information networks using tensor dynamics on battlespace manifolds, and integrating it with Lanchester-type attrition models.

To be able to compare our system with McLemore et al. [^{4} (see

To make our Red and Blue aircraft configurations more realistic, the identity 9-matrices (with the local feedback-loops) have been added to the Red and Blue adjacency matrices as:

Red : ( 0 0 0 0 1 1 1 1 1 0 0 0 0 1 1 1 1 1 0 0 0 0 1 1 1 1 1 0 0 0 0 1 1 1 1 1 1 1 1 1 0 0 0 0 0 1 1 1 1 0 0 0 0 0 1 1 1 1 0 0 0 0 0 1 1 1 1 0 0 0 0 0 1 1 1 1 0 0 0 0 0 ) noself-loops ⇒ ( 1 0 0 0 1 1 1 1 1 0 1 0 0 1 1 1 1 1 0 0 1 0 1 1 1 1 1 0 0 0 1 1 1 1 1 1 1 1 1 1 1 0 0 0 0 1 1 1 1 0 1 0 0 0 1 1 1 1 0 0 1 0 0 1 1 1 1 0 0 0 1 0 1 1 1 1 0 0 0 0 1 ) ≡ A b a includinglocalself-loops , (4)

Blue : ( 0 0 0 1 1 1 1 1 1 0 0 0 1 1 1 1 1 1 0 0 0 1 1 1 1 1 1 1 1 1 0 0 0 1 1 1 1 1 1 0 0 0 1 1 1 1 1 1 0 0 0 1 1 1 1 1 1 1 1 1 0 0 0 1 1 1 1 1 1 0 0 0 1 1 1 1 1 1 0 0 0 ) noself-loops ⇒ ( 1 0 0 1 1 1 1 1 1 0 1 0 1 1 1 1 1 1 0 0 1 1 1 1 1 1 1 1 1 1 1 0 0 1 1 1 1 1 1 0 1 0 1 1 1 1 1 1 0 0 1 1 1 1 1 1 1 1 1 1 1 0 0 1 1 1 1 1 1 0 1 0 1 1 1 1 1 1 0 0 1 ) ≡ A b a includinglocalself-loops . (5)

The graphs for our Red and Blue forces, defined by adjacency matrices with local self loops, A b a and A b a , respectively, are presented, using default graph embeddings in Mathematica^{®}, in

As a “soft” introduction to dynamics of vector and tensor fields on battlespace manifolds, we define here the Combat-tensors, as the following matrix products (i.e., tensor contractions):

Red : N b a ( x , t ) = A b a T c b ( x , t ) , Blue: N b a ( x , t ) = A b a T b a ( x , t )

of the combat adjacency matrices A b a and A b a with the total Power tensors T b a ( x , t ) and T b a ( x , t ) . Each component of the Power tensors (9 × 9 of them on the battle-manifold M^{9}; see next section) is defined as a sum of the sigmoid spatiotemporal kink functions Tanh ( x , t ) and ArcTan ( x , t ) .

A set of all active and controllable degrees-of-freedom (DOFs) of an arbitrary complex system comprises the configuration manifold for that system (see [^{n}, it can be approximated with ℝ n , where n, the total number of DOFs, can be in millions (using computational framework outlined in the Appendix).

Complex warfighting dynamics on such battle-manifolds is naturally defined as an interplay of spatiotemporal vector and tensor fields flowing on them. For defining tensor expressions, we will use the abstract tensor notation with Einstein’s summation convention upon repeated indices; see [

On any battle-manifold M^{n} we can observe a dynamic interplay of various Actors, all defined by various vector and tensor fields, depending on their complexity.

Simpler Actors are formally defined as spatiotemporal vector-fields, v a = v a ( x , t ) , similar to velocities and forces from classical mechanics, or flow-velocities and vortices from fluid mechanics, or Hamiltonian vector-fields from generalized mechanics [

The main Actors on any battle-manifold M^{n} are the Red and Blue vector-fields, R a ( x , t ) and B a ( x , t ) , respectively, which represent either the Red-Blue populations, or any other power measure of the Red-Blue forces.

The main supporting Actor is the combat-tensor N b a ( x , t ) , defined earlier, which belongs to this category; N b a commutes with any other 2nd-order tensor field of the same covariance on the same battle-manifold M^{n} (e.g. T b a ( x , t ) , S b a ( x , t ) )―they can be added together as linear machines: M b a = N b a ± T b a ± S b a ± ⋯

All these tensor fields are spatiotemporal dynamical objects governed by tensor equations, similar to the elastic stress-strain relation: σ a b stress = E a b c d elasticity ε c d strain .

For simplicity, we assume the simple 9D battle-manifold M 9 ≈ ℝ 9 , coordinated by x = ( x 1 , ⋯ , x 9 ) , although all the calculations would equally work for any

manifold dimension (up to millions, using the computational framework outlined in the Appendix). We start the TCW modeling with the tensor Lanchester square law, which is the following vector/tensor generalization of Equation (2):

Red : R ˙ a = k A b a B b , Blue : B ˙ a = κ C b a R b , (6)

where the Red and Blue forces are now defined as vector-fields, R a = R a ( x , t ) and B a = B a ( x , t ) , and their effectiveness coefficients are denoted by k and κ. The tensor fields A b a = A b a ( x , t ) and C b a = C b a ( x , t ) represent the sum of their combat-tensors ( N b a and N b a ), their total power (or stress-energy) tensors ( S b a = S b a ( x , t ) and S b a = S b a ( x , t ) ), and the Red and Blue swarming matrices, R b a = R b a ( x , t ) and B b a = B b a ( x , t ) from McLemore et al. [

Red : A b a = N b a R-C2 ± S b a R-Power ± R b a R-McL ,

Blue : C b a = N b a B-C2 ± S b a B-Power ± B b a R-McL .

For example, on a 9D battle-manifold M^{9}, the basic tensor Lanchester Equation (6) expand as:

R ˙ 1 = k A 1 1 B 1 + k A 2 1 B 2 + k A 3 1 B 3 + k A 4 1 B 4 + k A 5 1 B 5 + k A 6 1 B 6 + k A 7 1 B 7 + k A 8 1 B 8 + k A 9 1 B 9 ,

R ˙ 2 = k A 1 2 B 1 + k A 2 2 B 2 + k A 3 2 B 3 + k A 4 2 B 4 + k A 5 2 B 5 + k A 6 2 B 6 + k A 7 2 B 7 + k A 8 2 B 8 + k A 9 2 B 9 ,

R ˙ 3 = k A 1 3 B 1 + k A 2 3 B 2 + k A 3 3 B 3 + k A 4 3 B 4 + k A 5 3 B 5 + k A 6 3 B 6 + k A 7 3 B 7 + k A 8 3 B 8 + k A 9 3 B 9 ,

R ˙ 4 = k A 1 4 B 1 + k A 2 4 B 2 + k A 3 4 B 3 + k A 4 4 B 4 + k A 5 4 B 5 + k A 6 4 B 6 + k A 7 4 B 7 + k A 8 4 B 8 + k A 9 4 B 9 ,

R ˙ 5 = k A 1 5 B 1 + k A 2 5 B 2 + k A 3 5 B 3 + k A 4 5 B 4 + k A 5 5 B 5 + k A 6 5 B 6 + k A 7 5 B 7 + k A 8 5 B 8 + k A 9 5 B 9 ,

R ˙ 6 = k A 1 6 B 1 + k A 2 6 B 2 + k A 3 6 B 3 + k A 4 6 B 4 + k A 5 6 B 5 + k A 6 6 B 6 + k A 7 6 B 7 + k A 8 6 B 8 + k A 9 6 B 9 ,

R ˙ 7 = k A 1 7 B 1 + k A 2 7 B 2 + k A 3 7 B 3 + k A 4 7 B 4 + k A 5 7 B 5 + k A 6 7 B 6 + k A 7 7 B 7 + k A 8 7 B 8 + k A 9 7 B 9 ,

R ˙ 8 = k A 1 8 B 1 + k A 2 8 B 2 + k A 3 8 B 3 + k A 4 8 B 4 + k A 5 8 B 5 + k A 6 8 B 6 + k A 7 8 B 7 + k A 8 8 B 8 + k A 9 8 B 9 ,

R ˙ 9 = k A 1 9 B 1 + k A 2 9 B 2 + k A 3 9 B 3 + k A 4 9 B 4 + k A 5 9 B 5 + k A 6 9 B 6 + k A 7 9 B 7 + k A 8 9 B 8 + k A 9 9 B 9 ,

B ˙ 1 = κ C 1 1 R 1 + κ C 2 1 R 2 + κ C 3 1 R 3 + κ C 4 1 R 4 + κ C 5 1 R 5 + κ C 6 1 R 6 + κ C 7 1 R 7 + κ C 8 1 R 8 + κ C 9 1 R 9 ,

B ˙ 2 = κ C 1 2 R 1 + κ C 2 2 R 2 + κ C 3 2 R 3 + κ C 4 2 R 4 + κ C 5 2 R 5 + κ C 6 2 R 6 + κ C 7 2 R 7 + κ C 8 2 R 8 + κ C 9 2 R 9 ,

B ˙ 3 = κ C 1 3 R 1 + κ C 2 3 R 2 + κ C 3 3 R 3 + κ C 4 3 R 4 + κ C 5 3 R 5 + κ C 6 3 R 6 + κ C 7 3 R 7 + κ C 8 3 R 8 + κ C 9 3 R 9 ,

B ˙ 4 = κ C 1 4 R 1 + κ C 2 4 R 2 + κ C 3 4 R 3 + κ C 4 4 R 4 + κ C 5 4 R 5 + κ C 6 4 R 6 + κ C 7 4 R 7 + κ C 8 4 R 8 + κ C 9 4 R 9 ,

B ˙ 5 = κ C 1 5 R 1 + κ C 2 5 R 2 + κ C 3 5 R 3 + κ C 4 5 R 4 + κ C 5 5 R 5 + κ C 6 5 R 6 + κ C 7 5 R 7 + κ C 8 5 R 8 + κ C 9 5 R 9 ,

B ˙ 6 = κ C 1 6 R 1 + κ C 2 6 R 2 + κ C 3 6 R 3 + κ C 4 6 R 4 + κ C 5 6 R 5 + κ C 6 6 R 6 + κ C 7 6 R 7 + κ C 8 6 R 8 + κ C 9 6 R 9 ,

B ˙ 7 = κ C 1 7 R 1 + κ C 2 7 R 2 + κ C 3 7 R 3 + κ C 4 7 R 4 + κ C 5 7 R 5 + κ C 6 7 R 6 + κ C 7 7 R 7 + κ C 8 7 R 8 + κ C 9 7 R 9 ,

B ˙ 8 = κ C 1 8 R 1 + κ C 2 8 R 2 + κ C 3 8 R 3 + κ C 4 8 R 4 + κ C 5 8 R 5 + κ C 6 8 R 6 + κ C 7 8 R 7 + κ C 8 8 R 8 + κ C 9 8 R 9 ,

B ˙ 9 = κ C 1 9 R 1 + κ C 2 9 R 2 + κ C 3 9 R 3 + κ C 4 9 R 4 + κ C 5 9 R 5 + κ C 6 9 R 6 + κ C 7 9 R 7 + κ C 8 9 R 8 + κ C 9 9 R 9 .

Similar expansions (though larger) hold for battle-manifolds of any dimensions and can be derived using the fast tensor package xTensor [

Assuming, for simplicity, the coordinate independence ( x = const ), both sets of expanded Lanchester equations represent sets of coupled nonlinear ODEs, which can be directly numerically solved, for any given Red and Blue initial conditions: R a ( 0 ) = R 0 a , B a ( 0 ) = B 0 a , using any adaptive Runge-Kutta ODE-solver (e.g. Cash-Karp, Fehlberg and Dormand-Prince integrators), or their corresponding manifold/Lie-group integrators (e.g. Runge-Kutta Munthe-Kaas).

In the general case of explicit coordinate dependence ( x = x ( t ) ), we would be actually dealing with the set of the first-order nonlinear PDEs, which would all require spatial discretization (e.g., using the Method of Lines, as implemented in Mathematica), after which the above mentioned ODE-solvers can be used again.

The same computational algorithms will apply, in both cases (ODEs and PDEs), also for the extended tensor Lanchester equations, formulated as follows.

Next, to include the Lanchester linear law Equation (3) into Equation (6), while keeping their covariance (so that each term represents a vector-field), we need to extend them with quadratic terms of the Lanchester unaimed-fire equations (linear law) as:^{5}

Red : R ˙ a = k A b a B b + k b F c d a b B c R d , Blue : B ˙ a = κ C b a R b + κ b G c d a b B c R d , (7)

where the fourth-order tensors F c d a b and G c d a b represent more complex, strategic, tactical and operational, Red and Blue capabilities, which can be defined either as the outer products of various matrices from [

F c d a b = strat c d a b Red ± tact c d a b Red ± oper c d a b Red , G c d a b = strat c d a b Blue ± tact c d a b Blue ± oper c d a b Blue . (8)

The basic Red and Blue tensor combat Equation (7) are implemented in Mathematica as the initial value problem for the following temporal vector-fields:

where the 2nd-order Red and Blue combat-tensors A a , b and C a , b are defined via sparse adjacency matrices (4) and (5) as:

and the 4th-order (strategic + tactical + operational) tensors F a , b , c , d and G a , b , c , d are defined as:

A sample simulation of the basic tensor combat Equation (7) is performed in Mathematica (see Figures 5-7) for 10 time units (to match the simulations given in [

The focus of our interpretation is the Red-Blue dynamics phase plot in

In contrast, the output from [

Based on this interpretation, we can see that our proposed tensor framework is capable of addressing the similar questions as those addressed by [

For the purpose of recasting the combat-dynamics Equations (7) into a control system, we will add to both Red and Blue forces simple-and-strong bang-bang (on-off) control inputs u a ( t ) and v a ( t ) of the form:

u a ( t ) = { 0 , for 0 ≤ t < τ 4 , 10 , for τ 4 ≤ t < τ 2 , 0 , for τ 2 ≤ t < 3 τ 4 , 10 , for 3 τ 4 ≤ t < τ , and v a ( t ) = { 0 , for 0 ≤ t < τ 4 , 0 , for τ 4 ≤ t < τ 2 , 10 , for τ 2 ≤ t < 3 τ 4 , 10 , for 3 τ 4 ≤ t < τ ,

where τ is the total simulation time (in our case τ = 10 time units, to match the scenario from [

In this way, we obtain the controlled tensor Red-Blue equations:

Red : R ˙ a = k A b a B b + k b F c d a b B c R d + u a , Blue : B ˙ a = κ C b a R b + κ b G c d a b B c R d + v a . (9)

The basic vector control inputs u a ( t ) and v a ( t ) are implemented in Mathematica (in the scalar form) as:

u [ t _ ] = 10 Piecewise [ { { 0 , 0 ≤ t < τ 4 } , { 1 , τ 4 ≤ t < τ 2 } , { 0 , τ 2 ≤ t < 3 τ 4 } , { 1 , 3 τ 4 ≤ t < τ } } ] ,

v [ t _ ] = 10 Piecewise [ { { 0 , 0 ≤ t < τ 4 } , { 0 , τ 4 ≤ t < τ 2 } , { 1 , τ 2 ≤ t < 3 τ 4 } , { 1 , 3 τ 4 ≤ t < τ } } ] ,

Which gives the implementation of the controlled Red-Blue Equations (9) as:

A sample simulation of the bang-bang controlled tensor combat Equation (9) is performed in Mathematica (see Figures 8-10) for 10 time units and random initial conditions.

From Figures 8-10 we can see that adding strong bang-bang control inputs to tensor combat equations completely changes the natural combat-dynamics behavior―control actions have the overall flattening effect. Even if the control inputs have lower amplitudes (e.g., 5 instead of 10) the outcome would be qualitatively similar: both the time-plots and the phase plot would be flattened out. From these computational observations we can infer that adding artificial control inputs to natural Red-Blue combat dynamics does not make real sense, because in reality the Red and Blue forces mutually control each other.

We have presented the basic development of the tensor-centric warfare (TCW),

as a tensor union and generalization of classical Lanchester combat equations and modern intention to orient the conduct of defence and military decision-making around functions that cross traditional hierarchical lines of command. Recognizing both the debates that continue about how best to do this and the limitations and weaknesses in military theories intended to inform and drive these developments, we have picked up the central feature of information and communications systems as a base infrastructure in future force design. The emphasis in our formalization lies in the possibility of better addressing the complexity and uncertainty inherent in war and battle, which, despite having been studied since the Military Enlightenment period, have continued to prove challenging to military thinking. In the sequel to this paper, presented in [

The authors are grateful to Dr. Tim McKay and Dr. Brandon Pincombe, Joint and Operations Analysis Division, Defence Science & Technology Group, Australia―for their constructive comments which have improved the quality of this paper.

Ivancevic, V., Pourbeik, P. and Reid, D. (2018) Tensor-Centric Warfare I: Tensor Lanchester Equations. Intelligent Control and Automation, 9, 11-29. https://doi.org/10.4236/ica.2018.92002

A network-computational framework, with networks/tensors of up to millions of nodes, can be developed using the publicly available Matlab^{®} toolbox supporting the cutting-edge topological research of brain cliques and cavities from computational neuroscience (the Blue Brain project [

All tensor expressions can be derived using the tensor package xTensor [