Dynamic Optimization Calculator

Model Inputs

Use the responsive input grid below. Large screens show three columns, smaller screens show two, and mobile shows one.

Planning Settings

Time horizon

Number of decision periods.

Initial state

Starting value of the state variable.

Discount factor

Use a value between 0 and 1.

State Grid

State minimum

Lower bound for state search.

State maximum

Upper bound for state search.

State step

Smaller steps improve precision and raise compute time.

Control Grid

Control minimum

Lowest allowed control value.

Control maximum

Highest allowed control value.

Control step

Smaller steps provide finer control choices.

Transition Equation

State coefficient (a)

Persistence of the current state.

Control coefficient (b)

Effect of control on next state.

Drift term (d)

Constant state shift each period.

Stage Objective Coefficients

State linear coefficient

Benefit gained from the state variable.

State quadratic penalty

Penalty for large state values.

Control linear cost

Linear cost of applying control.

Control quadratic penalty

Penalty for aggressive controls.

Interaction coefficient

Cross effect between state and control.

Terminal linear coefficient

End-of-horizon reward from final state.

Terminal quadratic penalty

Penalty for extreme final states.

Example Data Table

The sample values below match the default inputs in the calculator and give a practical finite-horizon optimization scenario.

Parameter	Example Value	Description
Time horizon	6	Six optimization periods.
Initial state	10	Starting system level.
Discount factor	0.95	Future value weighting.
State range	0 to 50	Allowed grid for state interpolation.
State step	1	Distance between state grid points.
Control range	0 to 15	Allowed control choices each period.
Control step	0.5	Distance between control options.
State transition	x(t+1) = 0.90x(t) + 1.20u(t) + 2.00	Evolution equation.
Stage objective	12x - 0.25x² - 5u - 0.60u² - 0.15xu	Per-period payoff rule.
Terminal objective	8x(T+1) - 0.10x(T+1)²	Final-state reward rule.

Formula Used

This calculator solves a finite-horizon dynamic optimization problem with one state variable and one control variable.

J = Σ[t=1 to T] δ^(t-1) [ s₁xₜ - s₂xₜ² - c₁uₜ - c₂uₜ² - γxₜuₜ ] + δ^T [ k₁xₜ₊₁ - k₂xₜ₊₁² ]

xₜ₊₁ = a·xₜ + b·uₜ + d

Vₜ(x) = max over u { g(x,u) + δ·Vₜ₊₁(a·x + b·u + d) }

Where: x is the state, u is the control, δ is the discount factor, and T is the horizon length.

Method: The calculator uses backward induction on a discrete state-control grid. When next-state values fall between grid points, linear interpolation estimates the continuation value.

How to Use This Calculator

Enter the horizon, initial state, and discount factor.
Set a state grid wide enough to cover expected trajectories.
Set a control grid that reflects feasible decisions.
Define the transition equation coefficients a, b, and d.
Enter payoff, penalty, and terminal coefficients.
Click Optimize Now to compute the best path.
Review summary metrics, the period-by-period path, and exports.

FAQs

1) What does this calculator optimize?

It finds the sequence of controls that maximizes total discounted value across a finite planning horizon, subject to the transition equation and chosen control limits.

2) Why do state minimum, maximum, and step matter?

They define the state grid used for dynamic programming. A wider grid captures more outcomes. A smaller step improves precision, but it also increases runtime and memory use.

3) What is the role of the discount factor?

The discount factor reduces the weight of future rewards. A value near 1 treats future periods as important. A lower value makes early rewards dominate the solution.

4) What happens if the next state leaves the grid?

Continuation values are clipped to the nearest grid boundary through interpolation. If this happens often, widen the state range so the optimization better reflects your system.

5) Can I model a cost minimization problem here?

Yes. Represent benefits as small or zero values and enter penalties as positive coefficients. Maximizing a negative net payoff works like minimizing total cost.

6) When should I reduce the step sizes?

Reduce steps when the solution changes sharply between nearby states or controls, or when you want a closer approximation to a continuous problem. Balance accuracy against speed.

7) Can controls or states be negative?

Yes, as long as your chosen ranges include negative values and the model interpretation supports them. The calculator works with positive or negative states and controls.

8) Why might results differ from an exact analytical solution?

This tool uses grid search and interpolation, which approximate a continuous optimization problem. Finer grids usually reduce the gap between numerical and exact solutions.