Inputs
Example data table
| Scenario | Model | x₀ | xtarget | T, N | Weights (q, r, qf) | Bounds (umin, umax) | Typical outcome |
|---|---|---|---|---|---|---|---|
| Stabilize | Linear (a = -0.6, b = 1) | 2 | 0 | 5, 100 | 1, 0.2, 10 | -5, 5 | Fast decay with moderate control energy |
| Bound-limited | Linear (a = 0.2, b = 1) | 3 | 0 | 6, 120 | 1, 0.05, 15 | -2, 2 | Smoother control saturates near limits early |
| Nonlinear correction | Nonlinear (a = -0.4, b = 1, c = 0.15) | 1.5 | 0 | 5, 150 | 1, 0.3, 12 | -4, 4 | Control counters quadratic drift to reach target |
These examples illustrate stable, bounded, and nonlinear cases using the same cost structure.
Formula used
This solver minimizes a quadratic performance index using a discretized optimal control approach:
The state is propagated with forward Euler:
The adjoint variable (costate) is integrated backward to compute gradients:
λk = λk+1 + Δt[ 2q(xk − xref) + λk+1 ∂f/∂x ]
The control is updated using projected gradient descent:
How to use this calculator
- Choose a dynamics model, then set coefficients a, b, and c if needed.
- Enter the initial state, target state, and optional reference state.
- Set the time horizon T and the number of steps N for discretization.
- Select weights q, r, and qf to balance tracking versus effort.
- Specify control bounds to model actuator limits or safety constraints.
- Pick a control initialization and tune the learning rate α.
- Press Solve Optimal Control, then review the results above.
- Use CSV or PDF buttons to export the computed trajectory.
Tip: If convergence is slow, increase N modestly, reduce α, or raise r to penalize aggressive control.
1) Why optimal control matters in physics
Many physics systems need time-varying inputs that balance accuracy and effort. Examples include thermal regulation, vibration damping, trajectory shaping, and field control. Optimal control formalizes this tradeoff with a cost function and searches for the best feasible control signal.
2) The dynamic model sets what “optimal” means
This calculator supports a linear model, x′ = a x + b u, and a nonlinear extension, x′ = a x + b u + c x². Linear dynamics suit small deviations and fast prototyping. Nonlinear terms approximate drift, self-interaction, or curvature that becomes important at larger amplitudes.
3) Cost function: tracking versus control energy
The running cost penalizes state error with q and control effort with r, then adds terminal accuracy using qf. Increase q for tighter tracking to xref. Increase r for smoother, lower-energy actuation. A practical starting set is q = 1, r = 0.1–1, and qf = 5–50.
4) Discretization: horizon and time step
The horizon T fixes the available maneuver time, while N sets Δt = T/N. Smaller Δt improves numerical stability and gives finer control resolution, but increases computation. For many problems, N = 100–500 works well; strongly nonlinear cases may need larger N and a smaller learning rate.
5) Adjoint gradients: efficient sensitivity evaluation
Instead of perturbing each control value, the adjoint (costate) propagates sensitivity backward from the terminal cost. This yields ∂J/∂u across the whole horizon with one backward pass, making iterative optimization practical even with hundreds or thousands of control variables.
6) Bounds model realistic actuator limits
Real controllers saturate due to voltage, power, torque, or safety limits. The solver enforces feasibility by clipping u to [umin, umax] after each gradient step. Tight bounds often create early saturation followed by a gradual relaxation as the state approaches the target.
7) Convergence and tuning guidelines
If the cost oscillates or increases, reduce α (for example 0.05 → 0.01). If control looks too sharp, increase r. If final error remains large, raise qf or extend T. For nonlinear cases, keep c physically consistent and avoid overly aggressive bounds.
8) Exported tables support analysis and reporting
CSV and PDF outputs include time, state, control, λ, and ∂J/∂u. Plot x(t) and u(t) to check stability and saturation, then iterate on weights, bounds, N, and α. This repeatable loop helps match numerical optimality to practical physics constraints in design and teaching.
1) What problem type does this solver target?
It targets single-state, bounded, continuous-time control problems discretized over a horizon, with quadratic running and terminal costs. It is ideal for teaching and rapid prototyping.
2) Why do I see slow convergence for some settings?
Slow convergence often occurs with large N, stiff dynamics, or high nonlinearity. Reduce α, increase r, or increase N gradually. Ensure the nonlinear coefficient c is physically reasonable.
3) How should I choose q, r, and qf?
Start with q = 1, r between 0.1 and 1, and qf between 5 and 50. Raise q for tighter tracking, raise r for smoother controls, and raise qf to hit the final target.
4) What does xref do?
xref is the reference state used in the running cost. If you want regulation, set xref equal to the desired operating point. For pure terminal targeting, keep xref near zero.
5) Why does the control saturate at the bounds?
Saturation means the unconstrained optimum requires more authority than allowed. Tight bounds force the solver to use maximum effort early. Loosen bounds, extend T, or increase r to reduce saturation.
6) Is Euler discretization accurate enough?
For many smooth problems it is adequate, especially with small Δt. If dynamics are stiff, increase N or reduce T per step. For high-accuracy research, consider higher-order integration in a dedicated code.
7) What is the meaning of λ and the gradient column?
λ is the costate that propagates sensitivity backward from the terminal cost. The gradient ∂J/∂u indicates how changing u at each step changes total cost. Values approaching zero suggest convergence.
Accurate controls emerge when models and constraints are consistent.