Proactively Manage Project Resources Without Authority
If team members are missing their deadlines you must first try to determine the cause. For example, if it is due to a lack of skills, this should be addressed through training or replacement resources. If it is because they do not fully understand the expectations you have, then you may have some changes to make as well.
Although the team members do not report to you functionally, their work on the project should still be input into their overall performance review. You can try to hold people accountable by making sure they understand that you will be providing performance feedback into their review. This should also be reiterated and agreed to by the functional managers
From a process management side, there are project management techniques and processes that should be utilized. First of all, if the availability and performance of the team is in doubt, you should raise this early as a project risk. As part of risk management, you need to put a proactive plan in place to make sure that this risk is addressed. When people miss their deadlines and your deadline is in jeopardy, you may need to raise an issue and perform issues management. During issues management, you again look for the cause of the problem and try to resolve it.
In addition, make sure your team members are communicating proactively with you. In many cases, it is not the fact that people miss their deadlines that gets you frustrated; it is that the team member does not tell you ahead of time. If the team member communicates proactively, you can see the problem beforehand while you still some ability to help. If he just misses the date and does not communicate, then he is not managing expectations as should be done. By the same token, the project manager needs to communicate proactively as well. Communicate well with your team and make sure they understand dates and expectations. Also communicate proactively with the functional managers and make sure they know when there are resource sharing issues or people performance issues.
Matrix management involves a complex and delicate balancing act between project managers and people managers. The project manager usually has limited people management authority in these situations. Even so, it is possible to complete your projects successfully. There are many project management processes and techniques that can help. Utilize them to raise risks and issues when needed. Also, make sure you utilize the project sponsor. The sponsor can help you generate urgency and focus, and can also have an impact on the functional managers to make sure that you have the resources you need to be successful.