The work, led by Shiqiang Zhang with Kebo Li and Yangang Liang of the National University of Defense Technology in Changsha, China, tackles a problem that classical guidance theory has largely sidestepped. Most optimal guidance laws assume the missile flies at roughly constant speed with a fixed guidance-system time constant, the lag between a commanded maneuver and the airframe actually executing it. During boost, that assumption collapses. Thrust changes the missile’s speed dramatically, drag reshapes the trajectory, and the time constant itself varies as dynamic pressure climbs. When the motor dies, the missile’s deceleration profile flips again. A guidance law tuned for one regime can behave badly in the other, and the discontinuity at the switch point can inject jerky, wasteful commands into the autopilot precisely when precision matters most.

Zhang and colleagues built their solution on a careful rethinking of the engagement geometry. Rather than describing the pursuit in a fixed inertial frame, they established the three-dimensional relative kinematics in a rotating line-of-sight coordinate system, one that spins to keep the target centered. This choice has a powerful consequence: a guidance law derived in the planar cross-section of that rotating frame retains full three-dimensional interception capability, because the out-of-plane dynamics are absorbed into the frame’s rotation. The mathematics of planar pursuit, which is far more tractable, can therefore be applied to a genuinely three-dimensional engagement without losing the target’s lateral motion.

On top of that geometry, the researchers formulated a linear time-varying guidance model. Instead of pretending the guidance-system time constant is fixed, they let it vary across the boost and coast phases, and they modeled the target-equivalent acceleration, the effective forcing the missile must counter, as changing between the two regimes as well. Crucially, the effects of aerodynamic drag are written directly into the guidance model rather than treated as an afterthought. Drag is not a small perturbation for a coasting missile; it is the dominant force reshaping the trajectory, and ignoring it biases the law’s predictions of where the engagement will end.

The heart of the paper is the derivation of the two-phase optimal guidance law itself. The authors posed a weighted effort cost, a measure of how much control energy the missile must spend to null the guidance error, and minimized it analytically using the Cauchy–Schwarz inequality, a classical tool that converts an optimization problem into a closed-form expression. The weighting function is not arbitrary. By construction, it enforces continuity of the guidance command across the boost–coast transition, meaning the commanded acceleration does not jump when the motor cuts out. That eliminates the need for ad hoc command smoothing, the engineering patch that designers typically bolt on afterward to keep autopilots from lurching at phase boundaries.

Accurate timing is the other pillar of the method. Optimal guidance laws depend on the time-to-go, the estimate of how many seconds remain until intercept, because the law shapes its commands over that remaining interval. A poor time-to-go estimate degrades every downstream calculation. Zhang’s team developed an improved time-to-go estimator that explicitly accounts for thrust dynamics, capturing how the missile’s speed evolves during boost rather than assuming a constant velocity. They also derived closed-form phase-equivalent drag averages, weighting the drag over each phase so that the piecewise-constant equivalent target acceleration preserves the true drag contribution to the terminal zero-effort miss. In the boost phase, the missile speed follows a hyperbolic-tangent predictor driven by thrust; in coast, a rational deceleration model driven by drag. Both yield analytic expressions that can be evaluated in real time.

The appendix mathematics reveals how carefully the authors balanced the two phases. Influence-energy integrals, which measure how past commands propagate to the terminal miss, are computed separately for boost and coast and then combined with phase weights tied to the time constants. In the boost-phase law, the coast-interval contribution enters the denominator through a weighted term; in the coast-phase law, the common factor cancels between numerator and denominator. This bookkeeping is what guarantees the command stays continuous across the switch by construction rather than by tuning. The authors also show how the equivalent target acceleration is assembled from the target’s own maneuver, the missile’s thrust-driven speed profile, and the phase drag averages, closing the model with expressions involving hyperbolic functions and logarithms that are all evaluable in closed form.

Simulation results reported in the paper show the payoff. The proposed law reduces miss distance compared with conventional approaches while maintaining robust performance across the boost–coast transition, and it does so in an energy-efficient manner, spending less control effort to achieve the same terminal accuracy. The authors attribute this to three design choices working together: the drag-aware guidance model, the thrust-aware time-to-go estimator, and the weighting function that enforces command continuity. Each alone would be an incremental improvement; together they address the fundamental mismatch between guidance laws designed for one flight regime and missiles that must fly through two.

Why does this matter beyond the mathematics? Modern air-to-air engagements are decided in fractions of a second against maneuvering targets, and every meter of miss distance is a meter the warhead must make up with blast and fragmentation. A guidance law that wastes control energy fighting its own discontinuities has less margin left for the terminal maneuver. Conversely, a law that flows smoothly from thrust to coast preserves that margin, potentially extending effective range or improving kill probability against evasive targets. The same principles apply beyond missiles, to any vehicle that must transition between powered and unpowered flight, from reusable launch stages to sounding rockets to hypersonic glide vehicles that boost and then coast through the upper atmosphere.

The research, supported by the National Natural Science Foundation of China under grants 12472359 and U2441205, joins a growing body of work on optimal guidance under realistic flight constraints. Earlier studies have tackled drag in impact-angle control, field-of-view constraints in three-dimensional pursuit, and cooperative guidance against maneuvering targets. What distinguishes this contribution is its focus on the switching moment itself, the instant when the physics changes character and most guidance laws are caught unprepared. By deriving the law from first principles with the transition built in, Zhang, Li, and Liang have shown that the messiest phase of missile flight can be tamed not with patches but with better mathematics, and that the closed-form elegance of classical optimal control still has something fresh to say about one of aerospace engineering’s oldest pursuit problems.

Subject of Research: Optimal guidance law design for the boost–coast switching phase of air-to-air missile interception

Article Title: Three-Dimensional Optimal Guidance Law for Boost–Coast Switching Phase of the Air-to-Air Missile

Article References: Zhang, S., Li, K., & Liang, Y. (2026). Three-Dimensional Optimal Guidance Law for Boost–Coast Switching Phase of the Air-to-Air Missile. International Journal of Aeronautical and Space Sciences. https://doi.org/10.1007/s42405-026-01312-6

Image Credits: AI Generated

DOI: 10.1007/s42405-026-01312-6

Keywords: optimal guidance law, boost–coast switching, air-to-air missile, Cauchy–Schwarz inequality, line-of-sight rotating coordinate system, linear time-varying systems, time-to-go estimation, aerodynamic drag, Three-Dimensional, Optimal, Guidance, Boost