## 1 Introduction  
Interception of ballistic missiles has been a research topic for years. Calculation of interception trajectories is usually performed off-line by using two-point boundary-value problem formulation. The paper [1] employs the proposed guidance law in real-world conditions, where interception trajectories are sought in \(R^{6}\), comprising of: three NED (north-east-down) coordinates, speed, path angle and heading of the interceptor. This usually calls for the use of optimization techniques to obtain the reference trajectories to follow. In [2], there are four different guidance laws considered, showing that the interception of the missile is possible if one has perfect measurements and accurate estimation of the target maneuvers. To obtain the optimal interception point, one can use multiple approaches, ranging from classical variational calculus as in this paper, or others, such as metaheuristics, e.g., a genetic algorithm, see [3], which is used there to establish the optimal interception point, minimizing certain functional. The functional may refer to flight time of the intercepting missile, or distance between the interceptor and the ballistic missile. Guidance of an intercepting rocket is a problem which has multiple solutions. One can distinguish passive and active guidance approaches [4]. Several types of active guidance methods can be distinguished:  
* line of sight guidance (also called a two-point guidance), where the intercepting rocket always points to the target,
* deviated pursuit, where the intercepting missile points ahead of the target,
* proportional navigation, where the line of sight angle of the target is estimated,
* optimal guidance by minimizing a certain functional. Among the last, one can consider many approaches, such as LQR [5] embedded into variational calculus framework, LQI (LQ integrated) control law [6], applied to linearized systems, predictive control and nonlinear predictive control, which fall into the optimization strand [7], or performance-based approaches to PID tuning, [8]. This paper adopts the optimization technique to generate the optimal open-loop control signal to reach the desired point minimizing a certain performance index. A more comprehensive view on a general framework of a ballistic missile defense systems, comprising of a launcher and a radar, please refer to [9]. Another study of tracking a ballistic missile can also be found in [10]. Optimization techniques have been developed for ages--it was in 17 th century when such mathematicians as [PERSON], [PERSON], [PERSON] of d'Hospital pushed their horizons further. They are required in production processes, business/management, or in military applications, since everybody is interested in gathering as much as possible, at the least expense, in the shortest time, etc. The aim of this paper is to use this approach in the optimal interception problem of a ballistic rocket with a known trajectory, using the intercepting missile. As has been already remarked, an optimization approach is attractive in many areas, and can include, e.g., quadratic cost function in an unbounded problem, leading to an easy solution. In real world applications, however, constraints are ubiquitous, and need to be taken into consideration during the optimization process. In this work, optimization is based on variational calculus that could be, e.g., presented as the problem of finding, among all the continuous functions \(x(t)\), the one that satisfies terminal conditions [11]:  
\[x(t_{0})=x_{0}\,,\qquad x(t_{f})=x_{f}\,, \tag{1}\]  
achieving local extremum of the functional  
\[J(x)=\int_{t_{0}}^{t_{f}}f(f(t,\,x(t),\,\dot{x}(t))\,dt\,. \tag{2}\]  
The latter task has fixed terminal conditions, and the necessary condition of existence of such an extremum is satisfying the Euler-Lagrange (E-L) equation  
\[\frac{\partial f}{\partial x}-\frac{d}{dt}\left(\frac{\partial f}{\partial\dot {x}}\right)=0\,. \tag{3}\]  
From now on, whenever applicable without misunderstandings, the time indexes will be omitted.