A Geometric--Control Abstraction of One--on--One Air Combat Maneuvering

A Geometric–Control Abstraction of One–on–One Air Combat Maneuvering Shiqiao Luo December 5, 2025 Abstract We develop a rigorous mathematical framework showing that a broad class of dynamical models of one–on–one air combat maneuvering (ACM) admits a kinematic core: a purely geometric–control structure describing admissible trajectories and terminal outcome conditions, independent of the internal physical and engineering dynamics. The full ACM system is modeled as a controlled vector field on a smooth state manifold incorporating position, attitude, and internal variables. Under mild regularity assumptions, projection to a geometric configuration manifold induces a closed convex control cone field, which determines a controlled path category and a controlled fundamental groupoid. Internal physical dynamics form a “cage,” modeled as a functor from a decorated path category to a category of internal states. A forgetful functor collapses this internal structure to yield the kinematic core, a quadruple (X, g, K, Outcome) consisting of a Riemannian airspace, a geometric control cone field, and a purely geometric outcome map. We prove that every sufficiently regular ACM system admits such a core that faithfully captures all physically realizable geometric trajectories. Several examples, including a constant–speed bounded–curvature model with a ray–based firing rule, illustrate the construction and its relevance for studying positional advantage and winning strategies at the level of “pure maneuvering.” Contents 1 Introduction and Motivation 2 1.1 Motivation 1: Separating geometry from physics . . . . . . . . . . . . . . . . . . . . 3 1.2 Motivation 2: A path–space viewpoint . . . . . . . . . . . . . . . . . . . . . . . . . . 3 1.3 Motivation 3: Foundations for positional advantage and winning regions . . . . . . . 3 1.4 Outline ........................................... 4 2 Full ACM system 4 2.1 Airspace and configuration manifolds . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 2.2 Single–aircraft state space and control system . . . . . . . . . . . . . . . . . . . . . . 4 2.3 Two–aircraftdynamics................................... 5 2.4 Geometricprojection.................................... 5 3 Geometric control structure 6 3.1 Geometric velocity sets and control cone field . . . . . . . . . . . . . . . . . . . . . . 6 3.2 Admissible geometric trajectories . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 3.3 Controlled path category PK(Y) ............................. 7 3.4 Controlled fundamental groupoid ΠK 1(Y)......................... 7 1 4 Decorated path category and internal dynamics 8 4.1 Internalstatebundle.................................... 8 4.2 Decoratedpathcategory.................................. 8 4.3 Internaldynamicsfunctor ................................. 9 4.4 Forgetfulfunctor ...................................... 9 5 Kinematic core and outcome 10 5.1 Outcomemap........................................ 10 5.2 ACMkinematiccore.................................... 10 5.3 Full ACM system and kinematic core: existence theorem . . . . . . . . . . . . . . . . 10 6 Examples 11 6.1 Constant speed with bounded curvature in R3...................... 11 6.2 Ray–based firing outcome rule . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 6.3 Planarsimplification.................................... 12 7 Discussion and further directions 13 1 Introduction and Motivation Air combat maneuvering (ACM), particularly in the one–on–one (1v1) close–in regime, is governed by a complex interplay of aerodynamics, propulsion, flight–control laws, sensor models, and pilot physiology. Realistic engineering simulations operate on high–dimensional state spaces and incorporate detailed models of lift, drag, thrust, structural limits, and avionics. Yet practitioners often reason about ACM using much more abstract concepts: relative position, angle, energy, and “lines” or trajectories in space. From the perspective of geometry and control, it is natural to ask whether there is a mathematically precise sense in which ACM admits a universal maneuvering structure that depends only on how aircraft can move through space and how winning conditions are defined, but not on the specific details of aerodynamic coefficients or engine dynamics. Informally: Can we systematically “lock all engineering details in a cage” and obtain a canonical core that only sees geometry and control constraints, such that ACM advantage and winning strategies can be analyzed at this core level? The goal of this paper is to answer this question in the affirmative for a broad class of ACM models. We construct a two–layer abstraction: (i) An inner layer: a full ACM dynamical system on a smooth state manifold Z, with a controlled vector field incorporating all physical and engineering effects. (ii) An outer layer: a kinematic core A= (X, g, K, Outcome), where: •(X, g) is a smooth Riemannian manifold modeling the airspace; •Kis a closed convex control cone field on a configuration manifold Yderived from X; •Outcome : Y→ R (with Ra finite set of outcomes) is a purely geometric rule specifying termination conditions such as “A is pointing at B,” “B is pointing at A,” or “both are in mutual kill positions.” 2 The inner layer induces, via projection, a geometric control cone field Kon a configuration manifold that only remembers aircraft positions and headings. This cone field determines a category of admissible geometric paths and a controlled fundamental groupoid. The internal dynamic details—aerodynamic states, fuel levels, engine parameters, and so on—form a decorated structure over these paths, which we model as a functor from a decorated path category to a category of internal state spaces. A forgetful functor then discards this decoration, leaving only the kinematic core. There are several motivations for this abstraction. 1.1 Motivation 1: Separating geometry from physics In many strategic questions about ACM—for example, whether one pilot can guarantee moving into a position where the opponent enters some firing envelope—the underlying concern is geometric reachability under control constraints, not the exact form of aerodynamic forces. If different aircraft types are “similar enough” in terms of the sets of instantaneous velocities they can achieve at a given geometric configuration, one expects that their ACM behavior shares a common geometric structure. The present framework makes this intuition precise: the geometric control cone field Ksummarizes which infinitesimal motions are available at each geometric configuration. Different physical models that induce the same K—or Kin the same “equivalence class”—share the same kinematic core. 1.2 Motivation 2: A path–space viewpoint ACM is naturally path–based: one cares about trajectories in space and how they intersect certain “lines of fire.” Mathematically, such questions are well–suited to a path–space and category– theoretic viewpoint. We exploit this by introducing: •a controlled path category whose objects are configurations and whose morphisms are admissible paths; •a decorated path category that adds internal state evolution to each path; •a forgetful functor expressing the passage from full physics to pure geometry. This provides a rigorous realization of the idea that “all internal dynamics are decorations on geometric paths.” 1.3 Motivation 3: Foundations for positional advantage and winning regions Once the kinematic core is defined, one can study positional advantage, winning regions, and strategy in ACM by analyzing the controlled path category and outcome map, without explicit reference to internal states. For example, one may define winning regions as subsets of the configuration space from which a player has a strategy to force the outcome “A wins.” The geometry of these regions and their boundaries is determined by (X, g, K, Outcome) alone. Although this paper focuses on the structural decomposition and existence of the core, the framework is intended as a foundation for future work on geometric and game–theoretic analysis of ACM. 3 1.4 Outline Section 2 defines the full ACM system on a smooth state manifold and states regularity assumptions. Section 3 introduces the induced geometric control cone field and controlled path category. Section 4 develops the decorated path category, internal dynamics functor, and the forgetful functor that discards internal structure. Section 5 defines the ACM kinematic core and states the main existence theorem. Section 6 presents illustrative examples, including a constant–speed bounded–curvature model with a ray–based firing outcome. Section 7 discusses directions for further work. Throughout, all manifolds are assumed to be smooth (C∞), paracompact, second countable, and Hausdorff unless stated otherwise. 2 Full ACM system 2.1 Airspace and configuration manifolds Assumption 2.1. The airspace is modeled by a connected smooth 3–dimensional manifold X, which is paracompact, second countable, and Hausdorff, endowed with a smooth Riemannian metric g. The Riemannian structure provides distance and angle notions needed to define concepts such as “forward direction” and geometric firing conditions. Assumption 2.2. The attitude manifold His a compact smooth manifold (e.g. S2or SO(3)) representing the oriented direction or full attitude of an aircraft. For two aircraft Aand B, we define the geometric configuration manifold Y:= (X×H)A×(X×H)B. An element y∈Yis written y= (xA, hA;xB, hB). 2.2 Single–aircraft state space and control system Assumption 2.3. The internal state space Ξ is a finite–dimensional smooth manifold representing internal engineering variables (airspeed, engine settings, fuel state, internal flight–control states, etc.). Definition 2.4. The single–aircraft state manifold is S:= X×H×Ξ. We write s= (x, h, ξ)∈S. Assumption 2.5. The control input set Uis a nonempty compact metric space. Definition 2.6. A single–aircraft controlled dynamics is a map f:S×U→TS, satisfying the following regularity properties. Assumption 2.7 (Single–aircraft controlled regularity).The map f:S×U→TS satisfies: 4 (i) fis continuous in (s, u) and locally Lipschitz in suniformly in u: for every compact K⊂S there exists LK>0 such that ∥f(s1, u)−f(s2, u)∥ ≤ LK∥s1−s2∥,∀s1, s2∈K, ∀u∈U. (ii) For each measurable control u: [0, T]→Uand initial state s0∈S, the Carath´eodory system ˙s(t)=f(s(t), u(t)), s(0) = s0 admits a unique absolutely continuous solution on some maximal interval [0, Ts0,u). The norm ∥ · ∥ is induced by any fixed auxiliary Riemannian metric on S; all such choices are equivalent on compact sets. 2.3 Two–aircraft dynamics Definition 2.8. The full two–aircraft state manifold is Z:= SA×SB∼ =S×S. A state z∈Zcan be written z= (sA, sB)=(xA, hA, ξA;xB, hB, ξB). Definition 2.9. A full ACM system is a controlled vector field FZ:Z×U×U→TZ, so that the controlled dynamics are ˙z(t)=FZ(z(t), uA(t), uB(t)), uA, uB: [0, T]→U. Assumption 2.10 (Full ACM regularity).The map FZsatisfies: (i) FZis continuous in (z, uA, uB) and locally Lipschitz in zuniformly in (uA, uB). (ii) For each measurable controls (uA, uB) : [0, T]→U×Uand each z0∈Z, the Carath´eodory system admits a unique absolutely continuous solution on some maximal interval. A canonical example is FZ(z, uA, uB) = f(sA, uA), f(sB, uB), but our construction allows more general couplings as long as Assumption 2.10 holds. 2.4 Geometric projection Definition 2.11. The geometric projection π:Z→Yis π(xA, hA, ξA;xB, hB, ξB) = (xA, hA;xB, hB). Thus πdiscards internal variables and retains only position and attitude for each aircraft. 5 3 Geometric control structure 3.1 Geometric velocity sets and control cone field Definition 3.1. For each y∈Y, define the attainable geometric velocity set: E(y) := v∈TyY| ∃z∈Z, π(z)=y, ∃(uA, uB)∈U2,dπz(FZ(z, uA, uB))=v.(3.1) Assumption 3.2 (Geometric attainability regularity).For every y∈Y: (i) E(y) is nonempty and bounded in TyY; (ii) the set–valued map y7→ E(y) is upper semicontinuous with compact values: if yn→yand vn∈E(yn) with vn→v, then v∈E(y). Define the geometric control cone field: Definition 3.3. For each y∈Y, define K(y) := co E(y), the closed convex hull of E(y)inTyY. Lemma 3.4. Under Assumption 3.2, for every y∈Y, the set K(y)is nonempty, compact, and convex. Moreover, the set–valued map y7→ K(y)is upper semicontinuous in the Hausdorff topology. Proof. Nonemptiness and boundedness of E(y) imply that the convex hull is bounded and nonempty. Its closure is then compact and convex. Upper semicontinuity of Etogether with continuity of the closure and convex hull operations in the Hausdorff topology implies upper semicontinuity of K; see, e.g., [1, Ch. 5]. 3.2 Admissible geometric trajectories Definition 3.5. Let I= [0, T] with T > 0. An absolutely continuous curve γ:I→Yis called admissible (or K–admissible) if ˙γ(t)∈K(γ(t)) for almost every t∈I. The next proposition shows that projections of ACM trajectories are admissible. Proposition 3.6. Assume 2.10 and 3.2. Let z: [0, T ]→Zbe a solution of the full ACM dynamics with measurable controls uA, uB. Then γ=π◦z: [0, T]→Yis admissible in the sense of Definition 3.5. Proof. Since zis absolutely continuous, so is γ=π◦z. For almost every t, the derivative exists and ˙γ(t)=dπz(t)FZ(z(t), uA(t), uB(t)). By Definition (3.1), ˙γ(t)∈E(γ(t)) ⊂K(γ(t)). Thus γis admissible. A converse—that every admissible trajectory arises as a projection of a full ACM trajectory— requires additional technical conditions (e.g. viability in the fibers of πand measurable selection of controls). We return to this in Theorem 5.5. 6 3.3 Controlled path category PK(Y) We now formalize the path–space structure induced by K. Definition 3.7 (Equivalence of admissible curves).Two admissible curves γ1, γ2: [0,1] →Ywith γ1(0) = γ2(0) and γ1(1) = γ2(1) are said to be equivalent by time reparametrization if there exists an orientation–preserving C1diffeomorphism ϕ: [0,1] →[0,1] with ϕ(0) = 0, ϕ(1) = 1 such that γ2=γ1◦ϕ. We denote the equivalence class of γby [γ]. Definition 3.8 (Controlled path category).The controlled path category PK(Y) is defined as follows: •Objects: points y∈Y. •Morphisms: for y0, y1∈Y, the morphism set HomPK(Y)(y0, y1) consists of all equivalence classes [γ] of admissible curves γ: [0,1] →Ywith γ(0) = y0and γ(1) = y1under the relation in Definition 3.7. •Composition: given [γ1]:y0→y1and [γ2]:y1→y2, their composition is the equivalence class of the concatenated path (γ2∗γ1)(t) = (γ1(2t), t ∈[0,1/2], γ2(2t−1), t ∈[1/2,1], followed by reparametrization. •Identities: for each y, the constant curve γ(t)≡ydefines the identity idy. One checks that this defines a small category: associativity holds up to reparametrization, and identities behave as expected. 3.4 Controlled fundamental groupoid ΠK 1(Y) We refine the equivalence relation by introducing controlled homotopy. Definition 3.9 (Controlled homotopy).Let γ0, γ1: [0,1] →Ybe admissible curves with γ0(0) = γ1(0) and γ0(1) = γ1(1). They are K–homotopic if there exists a continuous map H: [0,1] ×[0,1] →Y such that (i) H(0, t) = γ0(t) and H(1, t)=γ1(t) for all t; (ii) H(s, 0) and H(s, 1) are constant in s; (iii) for each fixed s, the curve t7→ H(s, t) is admissible. This is an equivalence relation on admissible curves with fixed endpoints. Definition 3.10 (Controlled fundamental groupoid).The controlled fundamental groupoid ΠK 1(Y) is the category with: 7 •Objects: points y∈Y; •Morphisms: for y0, y1∈Y, morphisms are K–homotopy classes of admissible curves from y0 to y1. Composition is induced from PK(Y). Every morphism is invertible (by reversion of admissible paths), so ΠK 1(Y) is a groupoid. This groupoid is a controlled analogue of the usual fundamental groupoid, with admissibility constrained by the cone field K. 4 Decorated path category and internal dynamics We now formalize internal dynamics as “decorations” on geometric paths. 4.1 Internal state bundle Assumption 4.1 (Internal state bundle).There exists a smooth manifold Eand smooth maps ρ:Z→E, πE:E→Y such that: (i) π=πE◦ρ; (ii) πE:E→Yis a smooth surjective submersion with typical fiber F(so Eis a smooth fiber bundle over Y); (iii) for each y∈Y, the fiber Ey:= π−1 E(y) represents the internal states consistent with geometric configuration y. In the simplest case, one may take E=Zand ρ= idZ, and πE=π, but it is sometimes convenient to factor out internal redundancies. 4.2 Decorated path category Definition 4.2 (Decorated path category).The decorated path category Pdecor K(Y) is defined as: •Objects: pairs (y, ξ) with y∈Yand ξ∈Ey. •Morphisms: a morphism (y0, ξ0)→(y1, ξ1) is a pair (γ, Tγ) where: (a) γ: [0,1] →Yis an admissible curve with γ(0) = y0,γ(1) = y1; (b) Tγ:Ey0→Ey1is a smooth map representing the internal state evolution along γinduced by the projected dynamics on E; (c) Tγ(ξ0) = ξ1. •Composition: given (γ1, Tγ1) : (y0, ξ0)→(y1, ξ1) and (γ2, Tγ2) : (y1, ξ1)→(y2, ξ2), their composition is (γ2, Tγ2)◦(γ1, Tγ1) := (γ2∗γ1, Tγ2◦Tγ1), where γ2∗γ1denotes path concatenation. •Identities: (idy,idEy) at each (y, ξ). The existence and well–posedness of Tγfor admissible γfollow from Assumptions 2.10 and 4.1: the full ACM dynamics lift admissible curves in Yto integral curves in E. 8 4.3 Internal dynamics functor Let Cdenote the category whose objects are the fibers Eyand whose morphisms are smooth maps between these fibers arising as internal evolution operators along admissible curves. Definition 4.3 (Internal dynamics functor).The internal dynamics functor F:PK(Y)→ C is defined by: F(y) = Ey, F([γ])=Tγ, where [γ] denotes the morphism class in PK(Y) and Tγis the corresponding internal evolution operator. One checks that Fis well–defined: if γ1and γ2are equivalent under reparametrization, the induced evolution operators coincide. Composition and identities are preserved: F([γ2]◦[γ1])=F([γ2∗γ1])=Tγ2∗γ1=Tγ2◦Tγ1=F([γ2]) ◦F([γ1]), and F(idy) = idEy. If internal evolution depends only on K–homotopy classes of admissible curves (which may hold under additional conditions), Ffactors through ΠK 1(Y), yielding a functor ¯ F: ΠK 1(Y)→ C. 4.4 Forgetful functor Definition 4.4 (Forgetful functor).The forgetful functor U:Pdecor K(Y)→ PK(Y) is defined on: •Objects: U(y, ξ)=y; •Morphisms: U(γ, Tγ)=[γ]. Lemma 4.5. The functor U:Pdecor K(Y)→ PK(Y)is faithful. Proof. If (γ1, Tγ1) and (γ2, Tγ2) are morphisms with the same source and target and U(γ1, Tγ1) = U(γ2, Tγ2), then γ1and γ2are equivalent under reparametrization. The internal evolution operators Tγ1, Tγ2are determined by the underlying dynamics and initial internal states; if the dynamics are deterministic, distinct internal evolutions correspond to distinct admissible curves or different initial conditions. Thus morphisms in Pdecor K(Y) are distinguished by their geometric component and internal evolution. In particular, Uis injective on morphism sets. The functor Uformalizes the idea that the full ACM dynamics on Zdefine a richer decorated structure, and the kinematic core is obtained by forgetting this decoration. 9