Multiple Shooting

janus::MultipleShooting provides a transcription method for optimal control problems that enforces continuity via high-accuracy numerical integration (using CasADi's integrator interface, e.g., CVODES or IDAS). It divides the time horizon into intervals with piecewise-constant controls and connects them with continuity constraints. This works in symbolic mode via the janus::Opti interface. The class lives in <janus/optimization/MultiShooting.hpp>.

Quick Start

#include <janus/janus.hpp>
 
janus::Opti opti;
janus::MultipleShooting ms(opti);
 
janus::MultiShootingOptions opts;
opts.n_intervals = 20;
opts.integrator = "cvodes";
opts.tol = 1e-6;
 
auto [x, u, tau] = ms.setup(3, 1, 0.0, 2.0, opts);
 
ms.set_dynamics([](const janus::SymbolicVector& x,
                    const janus::SymbolicVector& u,
                    const janus::SymbolicScalar& t) {
    janus::SymbolicVector dxdt(3);
    dxdt(0) = x(2) * janus::sin(u(0));
    dxdt(1) = -x(2) * janus::cos(u(0));
    dxdt(2) = 9.81 * janus::cos(u(0));
    return dxdt;
});
 
ms.add_continuity_constraints();
ms.set_initial_state(janus::NumericVector{{0.0, 10.0, 0.001}});
ms.set_final_state(0, 10.0);
ms.set_final_state(1, 5.0);
 
opti.minimize(/* objective */);
auto sol = opti.solve();

Core API

Method	Description
MultipleShooting(opti)	Construct with a janus::Opti instance
setup(n_states, n_controls, t0, tf, opts)	Create decision variables and time grid
set_dynamics(ode)	Set the ODE function: (x, u, t) -> dxdt
add_continuity_constraints()	Apply integrator-based continuity constraints
add_dynamics_constraints()	Unified alias for add_continuity_constraints()
set_initial_state(x0)	Set initial boundary condition
set_final_state(xf)	Set final boundary condition
n_intervals()	Number of shooting intervals
time_grid()	Normalized time grid [0, 1]

MultiShootingOptions exposes:

• n_intervals: number of shooting intervals
• integrator: integrator type ("cvodes", "rk", "idas")
• tol: integrator tolerance

Advantages over Direct Collocation

• High Accuracy: Uses variable-step/variable-order integrators instead of fixed-order polynomials.
• Stiffness Handling: Better suited for stiff systems where low-order schemes fail.
• Sparse Structure: Retains the block-sparse structure of the NLP.

Disadvantages

• Cost: Evaluating integrator sensitivities can be computationally expensive compared to polynomial defects.
• Initialization: Can be harder to initialize if dynamics are unstable.

Usage Patterns

Basic Workflow

janus::Opti opti;
janus::MultipleShooting ms(opti);
 
janus::MultiShootingOptions opts;
opts.n_intervals = 20;
opts.integrator = "cvodes";
opts.tol = 1e-6;
 
auto T = opti.variable(2.0); // Variable final time
auto [x, u, tau] = ms.setup(n_states, n_controls, 0.0, T, opts);
 
ms.set_dynamics([](const janus::SymbolicVector& x,
                    const janus::SymbolicVector& u,
                    const janus::SymbolicScalar& t) {
    return /* dxdt */;
});
 
ms.add_continuity_constraints();
ms.set_initial_state(x0);
ms.set_final_state(xf);
 
opti.minimize(T);
auto sol = opti.solve();

Unified Comparison Example

The file examples/optimization/transcription_comparison_demo.cpp runs and compares Direct Collocation, Multiple Shooting, Pseudospectral, and Birkhoff Pseudospectral on the same brachistochrone problem so you can compare solver behavior and NLP structure directly. It includes solve-time performance statistics and a convergence sweep across grid sizes.

When to Use

Use Case	Why
High-fidelity simulation	Integrator accuracy
Stiff ODEs	Adaptive implicit methods
Matching simulation results	Same integration scheme
Fewer decision variables	Accuracy without more nodes

See Also

• Transcription Methods Guide – Comparison of all four transcription methods
• Direct Collocation Guide – Polynomial defect-based transcription
• Pseudospectral Guide – Global polynomial transcription
• Birkhoff Pseudospectral Guide – Birkhoff-form transcription
• transcription_comparison_demo.cpp – Unified comparison example
• MultiShooting.hpp – API reference