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<p>This research develops and implements a comprehensive Python-based computational framework for solving optimal control problems in economic dynamics using Pontryagin’s Maximum Principle (PMP). We employ advanced numerical methods including shooting algorithms with adaptive ODE solving (RK45 method with automatic step-size control), robust optimization techniques with multiple starting points, and parallel computation for sensitivity analysis. The framework generates interactive 2D and 3D visualizations using Matplotlib and Plotly to reveal geometric properties of optimal trajectories. The simulations demonstrate exceptional numerical accuracy with convergence errors near zero (<InlineEquation ID="IEq1"> <EquationSource Format="TEX">\(&lt;10^{-15}\)</EquationSource> </InlineEquation>) for well-behaved scenarios while identifying computational challenges in extreme parameter regimes. Key quantitative findings include: (1) precise convergence times to steady state (52.5 periods for baseline), (2) geometric curvature metrics (<InlineEquation ID="IEq2"> <EquationSource Format="TEX">\(\kappa _{\max } = 0.187\)</EquationSource> </InlineEquation>, <InlineEquation ID="IEq3"> <EquationSource Format="TEX">\(\bar{\kappa } = 0.089\)</EquationSource> </InlineEquation>, curvature index <InlineEquation ID="IEq4"> <EquationSource Format="TEX">\(CI = 2.10\)</EquationSource> </InlineEquation>, <InlineEquation ID="IEq5"> <EquationSource Format="TEX">\(\tau _{\text {total}} = 2.34\)</EquationSource> </InlineEquation> radians), and (3) sensitivity measures showing significant numerical challenges in high parameter regimes (errors up to 3614.14 for <InlineEquation ID="IEq6"> <EquationSource Format="TEX">\(\rho =0.04\)</EquationSource> </InlineEquation>). Our comparative analysis shows advantages over traditional shooting methods through adaptive step control and integrated visualization. This work establishes computational implementation and advanced geometric visualization as essential complements to theoretical analysis in economic dynamics, offering broad implications for sustainable economic research and policy applications.</p>

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A computational and geometric analysis of economic dynamics using Pontryagin’s maximum principle: advanced numerical simulations perspective

  • 期刊论文

  • 出版模式：

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  • 发表日期：

    2026年2月24日

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  • Walle Tilahun,
  • Bayenes Ayenew,
  • Sefinew Atinafu,
  • Silabat Enyew

摘要

This research develops and implements a comprehensive Python-based computational framework for solving optimal control problems in economic dynamics using Pontryagin’s Maximum Principle (PMP). We employ advanced numerical methods including shooting algorithms with adaptive ODE solving (RK45 method with automatic step-size control), robust optimization techniques with multiple starting points, and parallel computation for sensitivity analysis. The framework generates interactive 2D and 3D visualizations using Matplotlib and Plotly to reveal geometric properties of optimal trajectories. The simulations demonstrate exceptional numerical accuracy with convergence errors near zero ( \(<10^{-15}\) ) for well-behaved scenarios while identifying computational challenges in extreme parameter regimes. Key quantitative findings include: (1) precise convergence times to steady state (52.5 periods for baseline), (2) geometric curvature metrics ( \(\kappa _{\max } = 0.187\) , \(\bar{\kappa } = 0.089\) , curvature index \(CI = 2.10\) , \(\tau _{\text {total}} = 2.34\) radians), and (3) sensitivity measures showing significant numerical challenges in high parameter regimes (errors up to 3614.14 for \(\rho =0.04\) ). Our comparative analysis shows advantages over traditional shooting methods through adaptive step control and integrated visualization. This work establishes computational implementation and advanced geometric visualization as essential complements to theoretical analysis in economic dynamics, offering broad implications for sustainable economic research and policy applications.